Simple Vectorial Lie Algebras in Characteristic 2 and their Superizations

We overview the classifications of simple finite-dimensional modular Lie algebras. In characteristic 2, their list is wider than that in other characteristics; e.g., it contains desuperizations of modular analogs of complex simple vectorial Lie superalgebras. We consider odd parameters of deformations. For all 15 Weisfeiler gradings of the 5 exceptional families, and one Weisfeiler grading for each of 2 serial simple complex Lie superalgebras (with 2 exceptional subseries), we describe their characteristic-2 analogs - new simple Lie algebras. Descriptions of several of these analogs, and of their desuperizations, are far from obvious. One of the exceptional simple vectorial Lie algebras is a previously unknown deform (the result of a deformation) of the characteristic-2 version of the Lie algebra of divergence-free vector fields; this is a new simple Lie algebra with no analogs in characteristics distinct from 2. In characteristic 2, every simple Lie superalgebra can be obtained from a simple Lie algebra by one of the two methods described in arXiv:1407.1695. Most of the simple Lie superalgebras thus obtained from simple Lie algebras we describe here are new.


Notation and background
To make the text understandable for the uninitiated, we place the most basic facts before the Introduction which we have divided into two parts to make it more readable; for the same reason we divided the background to accommodate both students and experts. Statements proved directly, or by means of Mathematica-based SuperLie package [26], are called Claims.
We recall the basics and show how to modify familiar formulas in order to pass from C to fields of positive characteristic, especially characteristic 2. In some formulas given for p = 2, we retain notation convenient for comparison with the cases where p = 2.

Main points
1. We give an overview of the classification of simple finite-dimensional modular Lie algebras and Lie superalgebras over an algebraically closed field K of characteristic p > 0. We update the conjectures for various values of p > 0.
2. We use our results on classification of Lie superalgebras of vector fields with polynomial coefficients over C to describe their characteristic-p versions, especially, their desuperizations, for all 15 Weisfeiler gradings of all 5 exceptional simple vectorial Lie superalgebras, and for several serial ones, also exceptional in a sense.
3. One of the deforms 1 of the divergence-free Lie algebra svect(5; N ) which exists only if p = 2. It is one of the exceptional simple vectorial Lie algebras -a desuperization of an exceptional simple vectorial Lie superalgebra. This is the most unexpected result of this paper.

Generalities
As is now customary, we denote the elements of Z/2 by0 and1, to distinguish them from integers. For us, N := {1, 2, . . . }, as it used to be in the past, and still is in some countries; we set Z + := N ∪ {0}. The parity p of a non-zero element v of a Z/2-graded space V , called a superspace, is equal to i if and only if v ∈ V i . Any Z/2-graded algebra is called a superalgebra.
Hereafter K is an algebraically closed field of characteristic p > 0; usually, p = 2 and all Lie (super)algebras are finite-dimensional unless otherwise stated. Mostly (exceptions indicated), Π denotes the change of parity functor, i.e., tensoring by Π(Z).
The superization of most formulas of algebra is achieved via the following sign rule "If something of parity a is moved past something of parity b, the sign (−1) ab accrues. Formulas defined only on homogeneous elements are extended to arbitrary elements via linearity." Note that only even homomorphisms are considered as morphisms of superalgebras.
Observe that sometimes applying the Sign Rule requires some dexterity. For example, we have to distinguish between two versions both of which turn in the non-super case into one, called skew-or anti-commutativity, which are synonyms only in the non-super case; for two elements a and b of a superalgebra we call the following conditions Examples: the bracket in any Lie superalgebra is super anti-commutative; the anti-bracket {−, −} B.b. , see (2.5), being anti-commutative relative the parity in the Lie superalgebra is, however, super antiskew-commutative relative to the natural parity of generating functions.

Conventions and notation often used
In what follows, we assume that every supercommutative superalgebra is associative with 1; their morphisms send 1 to 1. We denote by c the center of a given Lie (super)algebra; both c(g) and cg := c ⊕ g denote a trivial central extension of g.
Let a b or b a denote the semi-direct sum of modules (algebras) in which a is a submodule (ideal).
Let d(g) := g KD, where D is an outer derivation of g; unless specified otherwise, D is the grading operator of g. For example, for d(o Π (2k)), we take D = diag(I k , 0 k ).
The symbol id (sometimes id n , id a|b ) denotes not only the identity operator (in the space of dimension n, resp. superspace of superdimension a|b), but also the tautological module V over the linear Lie superalgebra g ⊂ gl(V ); sometimes we write id g := V for clarity.
L D is the Lie derivative along the vector field D.

Definition of Lie superalgebras for p = 2, 3
The "naive" definition of Lie superalgebras for p = 2, 3 is obtained by applying the Sign Rule to anti-commutativity and Jacobi identities. To understand deformations with odd parameter, we need a more sophisticated approach using the functor of points. The multiplication in the Lie superalgebra will be called super-bracket or just bracket.

Definition of Lie superalgebras for p = 2
If p = 2, the antisymmetry condition for Lie algebra g0 should be replaced by an equivalent for p = 2, but otherwise stronger, alternating or antisymmetry condition [x, x] = 0 for any x ∈ g0.
If p = 2, a Lie superalgebra is a superspace g = g0 ⊕ g1 such that g0 is a Lie algebra, g1 is a g0module (made two-sided by symmetry), with a squaring x → x 2 and a bracket of odd elements, which are defined via a linear map s : S 2 (g1) −→ g0, where S 2 denotes the operator of raising to symmetric square, as follows: [x, y] := s(x ⊗ y + y ⊗ x) for any x, y ∈ g1.
The linearity of the g0-valued function s implies that (ax) 2 = a 2 x 2 for any x ∈ g1 and a ∈ K, and [x, y] is a bilinear form on g1 with values in g0.
The Jacobi identity involving odd elements takes the form of the following two conditions: for any x ∈ g1, y ∈ g0, x 2 , x = 0 for any x ∈ g1. (1.1) The (super)algebra satisfying only Jacobi identity, without any symmetry conditions, is called a Leibniz (super)algebra.
Over Z/2, the condition (1.1) must (for a reason, see [44]) be replaced with a more general one: [x, y]] for any x ∈ g1 and y ∈ g. (1.2) For any other ground field, this condition is equivalent to condition (1.1). More generally, for any Lie superalgebra g, since we want the Lie superalgebra der g of all derivations of g to be a Lie superalgebra, we have to add (to the Leibniz rule) the following condition on derivations (it becomes (1.2) for D = ad y ) D x 2 = [D(x), x] for odd elements x ∈ g1 and any D ∈ der g.
By an ideal i of a Lie superalgebra g, one always means i homogeneous with respect to parity, i.e., equal to i ∩ g0 i ∩ g1; for p = 2, the ideal should be closed with respect to squaring.
Recall that a given Lie (super)algebra g is said to be simple if dim g > 1 and g has no proper ideals; g is semisimple if its radical is zero; g is almost simple if it can be sandwiched (nonstrictly) between a simple Lie superalgebra s and the Lie superalgebra der(s) of derivations of s, i.e., s ⊂ g ⊂ der(s).
The definition of the derived of the Lie superalgebra g changes when p = 2: let g (0) := g and for any i ≥ 0, set An even linear mapping r : g −→ gl(V ) is said to be a representation of the Lie superalgebra g, and V is said to be a g-module if r([x, y]) = [r(x), r(y)] for any x, y ∈ g, r(x 2 ) = (r(x)) 2 for any x ∈ g1.

Definition of Lie superalgebras for p = 3
Since we are giving a review of the context for any p > 0, we have to note a peculiarity of p = 3, where the Jacobi identity for Lie superalgebras entails, additionally, that (1. 3) The super anti-commutative algebra satisfying the Jacobi identity, but not (1.3) is called a pre-Lie superalgebra.

Basics on the functor of points
In this subsection, we follow [59]; we advise the reader interested in subtleties that we, like most authors, do not dwell on, to read the Appendices to [60]. For a fixed object M and any object X of a category C, the association X −→ Hom C (X, M ) defines a functor F : C ; Sets. The idea is (A) To consider Hom C (X, M ) as the set of points of M , which is indeed the case for any set M if X is a point of M , and C = Sets; (B) Considering objects of the category C of sets endowed with a structure (of a group, algebra, module over a fixed algebra, topological space, etc.), and the morphisms in C being the maps of these sets preserving (exactly, or up to an equivalence, see [59,Section 1.16.3]) a certain structure (that of a group, or of an associative algebra, or of Lie algebra, etc.), as a model, we'd like to imitate these sets-with-a-structure by objects of another category.
For example, a Lie supergroup is any group in the category of supermanifolds, see [59]. Likewise, a Lie superalgebra is any Lie algebra in the category of linear supervarieties, see [44]. There we reformulate the naive definition of Lie superalgebras, which are Z/2-graded linear spaces with multiplication satisfying certain identities, in terms of supervarieties.

PBW-theorem for Lie superalgebras
In [23], an interesting description of conditions when the Poincaré-Birkhoff-Witt theorem for Lie superalgebras holds (or not) is offered for p > 0. Note, although we will not use this in this paper, that for Lie superalgebras understood "naively", the PBW theorem holds.

Deformations of the brackets
Let C be a supercommutative superalgebra, let Spec C be the affine super scheme.
Recall, see [62], where the non-super case is considered, that a deformation of a Lie superalgebra g over Spec C, is a Lie algebra G such that G g ⊗ C, as superspaces. The deformation is trivial if G g ⊗ C, as Lie superalgebras, not just as superspaces, and non-trivial otherwise.
Generally, the deforms of a Lie superalgebra g over K are Lie superalgebras G ⊗ I K, where I is any closed point in Spec C.
In particular, consider a deformation with an odd parameter τ . This is a Lie superalgebra G isomorphic to g ⊗ K[τ ] as a super space; if, moreover, G g ⊗ K[τ ] as a Lie superalgebra, i.e., then the deformation is considered trivial (and non-trivial otherwise). Observe that g ⊗ τ is not an ideal of G: the ideal should be a free K[τ ]-module.
Comment. Consider formal deformations over K [[τ ]]. If the formal series in τ converges in a domain D, we can evaluate τ for any τ ∈ D and -if dim g < ∞ -consider copies g τ , where τ ∈ D, of the same dimension as g. If the parameter is formal or odd, such an evaluation is possible only trivially: τ → 0.

Linear (matrix) Lie superalgebras
Certain basics of linear superalgebra are not well-known, or given wrongly in the literature; no harm in recalling about them.
The general linear Lie superalgebra of all supermatrices of size Size corresponding to linear operators in the superspace V = V0 ⊕ V1 over the ground field K is denoted by gl(Size), where Size = (p 1 , . . . , p |Size| ) is an ordered collection of parities of the basis vectors of V for which we take only vectors homogeneous with respect to parity and |Size| := dim V ; usually, for the standard (simplest from a certain point of view) format, gl(0, . . . ,0,1, . . . ,1) is abbreviated to gl(dim V0| dim V1). Any supermatrix from gl(Size) can be uniquely expressed as the sum of its even and odd parts; in the standard format this is the following block expression; on non-zero summands the parity is defined: The supertrace is the map gl(Size) −→ K, (X ij ) −→ (−1) p i (p(X)+1) X ii . Thus, in the standard format, str A B C D = tr A − tr D. Observe that for the Lie superalgebra gl C (p|q) over a supercommutative superalgebra C, i.e., for supermatrices with elements in C, we have So if C1 = 0, then on odd supermatrices the supertrace coincides with the trace. Since str [x, y] = 0, the subsuperspace of supertraceless matrices constitutes a Lie subsuperalgebra of gl(Size) called special linear and denoted sl(Size).

The queer version of gl(n)
There are at least two super versions of gl(n), not one; for reasons, see [52,Chapters 1 and 7]. The other version -q(n) -is called the queer Lie superalgebra and is defined as the one that preserves -if p = 2 -the complex structure given by an odd operator J, i.e., q(n) is the centralizer C(J) of J: It is clear that by a change of basis we can reduce J to the form (shape) J 2n in the standard format, and then q(n) takes the form (Over any algebraically closed field K, instead of J we can take any odd operator K such that K 2 = a id n|n , where a ∈ K × ; and the Lie superalgebras C(K) are isomorphic for distinct K; if p = 2, it is natural to select K 2 = id, and hence Π n|n := Π 2n = 0 1n 1n 0 can serve as the normal shape of K.) On q(n), the supertrace vanishes, but the queertrace is defined: qtr : (A, B) −→ tr B. Denote by sq(n) the Lie superalgebra of queertraceless matrices; set psq(n) := sq(n)/K1 2n .

The supermatrix of the dual operator (after [52])
Let F ∈ End C (V ). The passage from the matrix of F in a basis of V to the matrix of F * in the dual basis of V * is performed by means of the supertransposition which in the standard format is of the shape where M t is the transposed of the matrix M .

Lie superalgebras preserving bilinear forms
The supermatrices X ∈ gl(Size) such that constitute the Lie superalgebra aut(B) that preserves the bilinear form B on V whose Gram matrix B = (B ij ) is given by the formula In order to identify a bilinear form B(V, W ) with an operator, an element of Hom(V, W * ), the matrix B of the bilinear form B is defined in [52, Chapter 1] by equation (1.4), not by seemingly natural -but inappropriate for such an identification -formula (1.5) Moreover, the would-be definition (1.5) contradicts the manifest symmetry of the odd bilinear form qtr on q(n). To correctly define symmetry of bilinear forms, consider the upsetting of bilinear forms u : Bil(V, W ) −→ Bil(W, V ), see [52,Chapter 1], given by the formula

Notational convention
By abuse of notation we will often denote the bilinear form B by its Gram matrix B in a normal shape.

Analogs of polynomials for p > 0
Let C[x] := C[x 1 , . . . , x a ] denote the supercommutative superalgebra of polynomials in indeterminates x in their standard order ; i.e., let the first m indeterminates be even and the other n be odd (m + n = a). Among the bases of C[x] in which the structure constants are integers, the two bases are usually considered: the monomial basis and the basis of divided powers constructed as follows.
For any multi-index r = (r 1 , . . . , r a ), where r 1 , . . . , r m ∈ Z + and r m+1 , . . . , r a ∈ {0, 1}, we set Clearly, we have where r + s r These u (r i ) i form an "integer basis" (i.e., a basis in which all structure constants with respect to the product (1.7) are integers) of C[x].

Notational convention
In what follows, for clarity, we will write exponents of divided powers in parentheses.
Over any field K of characteristic p > 0, we consider the supercommutative superalgebra (now we do not have any elements x, only the u with multiplication given by formula (1.7) where N = (N 1 , . . . , N m ) is the shearing vector with N i ∈ Z + ∪ ∞ (we assume that p ∞ = ∞). Important particular cases of shearing vectors:  for all i and all k i such that 0 ≤ k i < N i if u i is even.

The (generalized) Cartan prolongation
Let g − = ⊕ −d≤i≤−1 g i be a nilpotent Z-graded Lie (super)algebra and g 0 a Lie sub(super)algebra of the Lie (super)algebra der 0 (g − ) of degree 0 derivations of g − . Recall that the graded Lie The maximal transitive Z-graded Lie (super)algebra whose non-positive part is g − ⊕ g 0 is called the (generalized) Cartan prolong of the pair (g − , g 0 ) and is denoted by (g − , g 0 ) * .
If p = 0, we can realize g − by elements of negative degree of vect(n|m; r) and g 0 by elements of 0th degree of vect(n|m; r) in a non-standard (see Section 2.4.1) grading of vect(n|m), where n|m = sdimg − . Then the Cartan prolong (g − , g 0 ) * := ⊕ k≥−d g k of the pair (g − , g 0 ) is obtained for any k > 0 by The above-described procedure is called generalized prolongation because the initial Cartan prolongation was defined for d = 1 only.

Partial Cartan prolongation involving positive components
The terms h i , where i > 2, are similarly defined. Set h i := g i for i ≤ 0 and call h * := ⊕h i the partial Cartan prolong involving positive components.
Examples. The Lie superalgebra vect(1|n; n) is a subalgebra of k(1|2n; n). The former is obtained as the Cartan prolong of the same nonpositive part as k(1|2n; n) and a submodule of k(1|2n; n) 1 . The simple exceptional superalgebra kas discovered in [64,65] is another example.

Vectorial Lie algebras and algebras of divided powers
The Cartan prolong of (g − , g 0 ), where g 0 acts faithfully on g − and sdim g − = m|n, can be embedded into the superalgebra of polynomial vector fields of m even and n odd indeterminates, i.e., into der C[x 1 , . . . , x a ] (where a = m + n, the first m indeterminates are even, and the rest are odd), see [66].
Over a field K of characteristic p > 0, if one tries to follow the recipe of Section 1.6 naively and use derivations of usual polynomials, instead of divided powers, it would not work. For example, let us consider the prolong (g − , g 0 ) * , where g − = g −1 , sdim g −1 = sdim g 0 = 1|0, and the action of g 0 on g −1 is non-trivial. It has the form ∞ i=−1 g i such that sdim g i = 1|0 and The corresponding prolong over C would be embedded isomorphically into der C[x] so that g i would be mapped into Span x i+1 ∂ x . Over K, the construction of embedding would fail for However, over K, Cartan prolongs can be embedded into the superalgebra of derivations of the algebra of divided powers. Let us first say a bit about these derivations.
Over C, consider the action of derivation ∂ x i of C[x 1 , . . . , x a ] in the basis of divided powers. It is given by (recall the definition (1.6) of u (r) ) ..,ra)) otherwise. (1.10) Since all the coefficients are integer, the map given by this formula is a derivation of K[m; N |n].
We will denote this map ∂ i := ∂ x i and call the maps ∂ 1 , . . . , ∂ a distinguished partial derivatives.
The general Lie algebra of vector fields consists of the following derivations expressed in terms of distinguished partial derivatives Note that if N = 1, then vect(m; N |n) is not the whole der K[m; N |n]. Maps ∂ p k i , where 1 ≤ i ≤ m and 1 ≤ k < N i , are also derivations of K[m; N |n], and a general derivation of K[m; N |n] has the form 2 The Lie superalgebra vect(m; N |n) and its subalgebras, are called vectorial Lie superalgebras (cf. with matrix or linear Lie superalgebras). Cartan prolongs can be embedded into vect(m; N |n); in particular, the above Cartan prolong would be isomorphic to vect(1; N ∞ |0), with g i corresponding to Span u (i+1) 1 ∂ 1 .

Notation, again
Hereafter, the symbol g(a|b) or g(a; N |b) will designate the vectorial Lie superalgebra with given name g realized by vector fields on the linear supermanifold K a|b (the one corresponding to the superspace K a|b , see [44]), and endowed with a W-grading, see Table (2.18). The standard grading is taken as a point of reference for regradings governed by the vector r of degrees, which often can be described by one number r that usually (for details, see [56,65]) is equal to the number of odd indeterminates of degree 0. The regraded Lie superalgebra is denoted by g(a|b; r). In the standard grading the parameter r is usually omitted, see Table (2.18) and tables in Section 25.4.
The module F of "functions" over vect(m; N |n) and its subalgebras (usually with the same negative part) is an analog of the tautological module V over gl(V ) and its subalgebras.

Names
The Lie algebra vect(1; N ) is called a Zassenhaus algebra. For p = 2 it is not simple. Observe that vect(1; N ) k(1; N ) (indeed, f ∂ x ←→ K f , see definition (2.2); clearly, ∂ x is the distinguished derivative with respect to the only indeterminate). The simple derived algebra vect (1) N ) is also called a Zassenhaus algebra causing confusion, while vect(1; 1) is lately called (even for p = 0) the Witt algebra in honor of Witt who was the first to study one of its modular incarnations, see Introduction to the first volume of [73].
In the old literature, vect(m; N ), like its version for p = 0, was called the general Lie algebra of Cartan type; lately, it is called the Jacobson-Witt algebra, whereas the name Witt algebra is reserved for the particular case vect(1; 1) for p > 2.

Traces and divergencies on vectorial Lie superalgebras
On any Lie (super)algebra g over a supercommutative superalgebra C, e.g., over a field C = K, a trace is any linear mapping tr : g −→ C such that tr g (1) = 0. (1.11) Now let g be a Z-graded vectorial Lie (super)algebra with g − := ⊕ i<0 g i generated by g −1 , and let tr be a trace on g 0 . Recall that any Z-grading of a given vectorial Lie (super)algebra is given by degrees of the indeterminates, so the space of functions F, see equation (1.9), is also Z-graded. The divergence div : g −→ F is a degree-preserving ad g −1 -invariant extension of the trace to the Cartan prolong; this extension should satisfy the following conditions, so div ∈ Z 1 (g; F), i.e., is a cocycle: We denote by Vol(u; N ) or simply Vol u := F * the vect(m; N |n)-module of volume forms dual to F over F. As an F-module, Vol u is generated by the volume element vol u = 1 * with fixed indeterminates ("coordinates") u which we often do not indicate. On the rank-1 F-module of weighted λ-densities Vol λ (m; N |n) with generator vol λ u over F, the vect(m; N |n)-action is given for any f ∈ F and D ∈ vect(m; N |n) by the Lie derivative The special Lie algebra sg := Ker div of divergence-free elements of g is the Cartan prolong of (g − , Ker tr | g 0 ). For example, svect(m; N |n) = (id sl(m|n) , sl(m|n)) * ,N .

Examples of several divergences
On vect(m; N |n), the explicit expression of the standard divergence is as follows div : (1.14) The supertrace restricted from gl vanishes on q, but there is an "indigenous" queer trace on q; analogously, the standard divergence (1.14) vanishes on certain Lie subsuperalgebras of vect(m; N |n) on which there might be defined an "indigenous" divergence. This happens, e.g., with k(2n + 1|2n + 2) and m(n) as will be shown later on.
If there are several traces on g 0 , and hence divergences on g = (g − , g 0 ) * ,N , there are several types of special subalgebras, and we need an individual name for each.
If g is a Lie super algebra, then the linear functional tr satisfying condition (1.11) is often called, for emphasis, super trace and denoted by str. If we were consistent, we should, accordingly, use the term super divergence but instead we drop the preface "super" in both cases.

Critical coordinates and unconstrained shearing vectors
The coordinate of the shearing vector N corresponding to an even indeterminate of the Z-graded vectorial Lie (super)algebra g is said to be critical if it cannot take an arbitrarily big value.
The shearing vector without any imposed restrictions on its coordinates is said to be unconstrained ; we denote it by N u . Let dim N be the number of coordinates of N , be the number of parameters N u depends on.
We established the (non)critical coordinates of the shearing vectors of the Z-graded vectorial Lie (super)algebra g with a computer's aid by explicitly computing the bases of the first several terms g i for i ≥ 0 without imposing any constraints on N .   We chose the central element z ∈ g 0 so that it acts on g i as i · id. The irreducible 1-dimensional module over the commutative Lie algebra spanned by z which acts as i · id is denoted by K[i].
The type m: The "odd" analog of k is associated with the following "odd" analog of hei(2n|m). Denote by ba(n) the antibracket Lie superalgebra (ba is Anti-Bracket read backwards). Its space is W ⊕ C · z, where W is an n|n-dimensional superspace endowed with a non-degenerate antisymmetric odd bilinear form B; the bracket in ba(n) is given by the following relations: z is odd and lies in the center; [v, w] = B(v, w) · z for any v, w ∈ W .
Given ba(n) and a subalgebra g of cpe(n), we call (ba(n), g) * the m-prolong of (W, g), where W is the tautological pe(n)-module.

Generating functions over C
A laconic way of describing k, m and their subalgebras is via generating functions.
The Buttin 3 bracket {−, −} B.b. , discovered by Schouten and initially known as the Schouten bracket, is very popular in physics under the name antibracket, see [24]. It is given by the formula In terms of the Poisson and Buttin brackets, respectively, the contact brackets are as follows: The Lie superalgebras of Hamiltonian vector fields (or Hamiltonian superalgebras) and their special subalgebras (defined only if n = 0) are The "odd" analogues of the Lie superalgebra of Hamiltonian fields are the Lie superalgebra of vector fields Le f introduced in [50], and its special subalgebra: It is not difficult to prove the following isomorphisms as Lie superalgebras with the brackets on the right-hand sides given by the above-described brackets k.b. (where K f and H f are involved) and m.b. (where M f and Le f are involved) We have
The vector fields D satisfying (2.8) for some function F D look differently for different characteristics: For p = 2, and also if p = 2 and n = 2k + 1, the fields D satisfying (2.8) have, for any f ∈ F, the following form (compare with (2.2)): For p = 2 and n = 2k+2, we cannot use formula (2.10) anymore (at least, not for arbitrary f ) since it contains 1 2 . In this case, the elements of the contact algebra are of the following three types, and their linear combinations, where k = k0 + k1: a) For any f ∈ F such that ∂f ∂x 0 = ∂f ∂x 2k+1 = 0, we have Observe that in (2.10) and (2.11) we can also take E := (2.12) For any g ∈ F, or equivalently, for any g ∈ F such that ∂g ∂x 2k+1 = 0, we set b1) For the pericontact Lie superalgebra m(n; N |n) the analog of the formula (2.9) takes the form where the parities of indeterminates are such that p(x i ) = p(x i+k ) +1; e.g., they are as follows even : x 1 , . . . , x k ; odd : x k+1 , . . . , x 2k , and x 0 .
Then, for any f, f 1 , g, g 1 ∈ F, we deduce , Proof is based on direct computations, see formulas (2.14).
In particular, for k(1; N |0), we have (unless a = b = 0) Lemma 2.2 (A helpful lemma). For any g ∈ F, see (2.12), we have g 2 ∈ K, see the expressions for (A g ) 2 and [A g , B g ] in equation (2.14).
Proof . Indeed, g = r g r x (r) , where r is a (2k + 1)-tuple of non-negative numbers and the sum runs over a set of such tuples, and g r ∈ K for all r. Then, i.e., the terms with r = s, are encountered 2 times, so what remains is 2ri ri x (2r) .  For p = 2, the element x 0 annihilates a subspace ann(x 0 )| g −1 of g −1 and acts as multiplication by 1 on both g −1 /(ann(x 0 )| g −1 ), and g −2 .
3) k(2k0 + 1; N |2k1 + 1) po(2k0;N |2k1) ⊕ F Π F , where N = (N 0 ,N ) and at least one of k0 and k1 is non-zero. These Lie superalgebras and their desuperizations are not simple, the ideal i is generated by the A g and B g ; recall (2.12). We have

Weisfeiler gradings
For vectorial Lie superalgebras, the invariant notion is filtration, not grading. In characteristic 0, the Weisfeiler filtrations were used in the description of the infinite-dimensional Lie (super)algebras L by selecting a maximal subalgebra L 0 of finite codimension; for the simple vectorial Lie algebra, there is only one such L 0 . (Dealing with finite-dimensional algebras for p > 0, we can confine ourselves to maximal subalgebras of least codimension, or "almost least".) Let L −1 be a minimal L 0 -invariant subspace strictly containing L 0 ; for i ≥ 1, set: We thus get a filtration: The d in (2.17) is called the depth of L, and of the associated Weisfeiler-graded Lie superalgebra We will for brevity say W-graded and W-filtered.
For the list of simple W-graded vectorial Lie superalgebras g = ⊕ −d≤i g i over C, see [55] reproduced in Tables 25.2 and 25.3.

The Z-gradings of vectorial Lie superalgebras
These gradings are defined by the vector r of degrees of the indeterminates, but this vector can be shortened for W-gradings to a number r, or a symbol, which we do not indicate for r = 0. Let the indeterminates t, p i , q j , and u be even, while τ , ξ i , η j , and θ be odd. Let the contact Lie superalgebra k(2n + 1|m) preserve the distribution given by the Pfaff equation where the form α 1 is given by (2.1). For the k series, let u = (t; p, q) be even indeterminates, the odd indeterminates being the θ (resp. θ, ξ, η), see (2.1). For the m series, the indeterminates in Table (2.18) are denoted as in formula (2.3), i.e., the q i even, the ξ i , and τ odd.
In Table (2.18), the "standard" gradings correspond to r = 0, they are marked by an asterisk ( * ). For r = 0, the codimension of L 0 is the smallest.

Divergence-free and traceless subalgebras
In this subsection, the ground field is any K for p = 2. The peculiarities of p = 2 are considered in Sections 2.9 and 2.10. Here we will not mention N if p > 2.

k series
Since the restriction of the standard divergence (1.14) to the subalgebra of degree 0 is (super)trace, and since the space g 0 /[g 0 , g 0 ], where g := k(2n + 1|m), is spanned by K t for (n, m) = (0, 2), it is easy to calculate that it follows that the divergence-free (relative the restriction of the divergence (1.14) to k(2n+1|m)) subalgebra of the contact Lie superalgebra either coincides with it for m = 2n+2 or is the Poisson superalgebra singled our by the condition ∂ t (f ) = 0. On k(2n + 1|2n + 2) there is its own, "indigenous" divergence K f → ∂ t (f ); it also singles out the Poisson superalgebra. This, however, is not the whole story: the case k(1|2) is exceptional.
The case of k(1|2). Let α 1 = dt + ξdη + ηdξ. Since k(1|2) 0 is commutative and 2-dimensional, there are 2 linearly independent traces on it: one -tr -is equal to 1 at t and vanishes at ξη, the other one -call it tr (2) -is equal to 1 at ξη and vanishes at t.
Clearly, the condition K 1 (f ) = 0 singles out the subalgebra k − ⊕ Kξη of k(1|2). In other words, the operator ∂ t = 1 2 K 1 in the adjoint representation is an analog of the divergence -the prolong of the trace on k 0 ; this analog is equal to 1 at t and vanishes at ξη.
The divergence-free condition div D = 0, where D ∈ g for a Z-graded vectorial Lie superalgebra g, should single out the complete prolong of (g − , s), where s = {g ∈ g 0 | tr g = 0}. Therefore, the condition that determines the divergence is X(div D) = div([X, D]) for any X ∈ g − . (2.20) Since g −1 generates the negative part, it suffices to require fulfillment of the condition (2.20) for any X ∈ g −1 . Therefore, we have to express the divergence not in terms of partial derivatives, but in terms of the operators commuting (not supercommuting) with g − (recall that in [66], the operators that span g − are denoted by X i , and the operators commuting with g − are denoted by Y i ).
The k(1|2)-module of weighted densities. Over contact Lie superalgebras k(2n + 1|m) it is natural to express the spaces of weighted densities in terms of the conformally preserved form α 1 . This recalculation is well-known for m = 0, where vol = α 1 ∧ (dα 1 ) n . The general case follows from equation (2.19): from the point of view of the k(2n + 1|m)-action Since the center of k(1|2) 0 is of dimension 2, the weights of the spaces of weighted densities have 2 parameters, not one: F a,b := Fα a 1 β b , where a, b ∈ K. Let β be the symbol of the class of the differential form dξ (or, equivalently, (dη) −1 )in the quotient space Ω 1 /Fα 1 of 1-forms. The Lie derivative acts as follows The space F a,b of weighted densities over k(1|2) is a rank-1 module generated by α a 1 β b over the algebra of functions F = F 0,0 .

m series, its simple subalgebras, and weighted densities
For the pericontact series, the situation is more interesting than that for contact series: the divergence-free subalgebra is simple and new (only as compared with the above-described algebras; it is known since ca 1978, see [1]).
Let p = 2. Since it follows that the divergence-free subalgebra of the pericontact superalgebra is In particular, The divergence-free vector fields from sle(n) are generated by harmonic functions, i.e., such that ∆(f ) = 0. Rank 1 over the algebra F modules F m a,b := Fα a 0 γ b , where a, b ∈ K, are generated by α a 0 γ b , where γ is a symbol of the class of differential forms (whose explicit expression is irrelevant, same as that of β, see equation (2.23)). The Lie derivative acts as follows: The divergence-free relative the standard divergence Lie superalgebras sle(n), sb(n) and svect(1|n) have traceless ideals sle (1) (n), sb (1) (n) and svect (1) (n) of codimension 1; they are defined from the exact sequences We denote the operator that singles out b λ (n) in m(n) as follows, cf. (1.12): Taking the explicit form of the divergence of M f into account, we get a ) the structure of b λ (n) differs from the other members of the parametric family: the following exact sequences single out simple Lie superalgebras (the quotient le(n) and ideals, the first derived subalgebras): Problem 2.6. The Lie superalgebras b λ (n) can be further deformed at certain points λ, see [58], where K = C; the Lie superalgebras of series h and le also have extra deformations. Describe the deformations of b λ (n; N ), as well as h and le for all p > 0.
2.6 Passage from p = 0 to p > 0 Here we have collected answers to several questions that stunned us while we were writing this paper. We hope that even the simplest of these answers will help the reader familiar with representations of Lie algebra over C, but with no experience of working with characteristic p > 0. For p = 2, several of our definitions are new, see Sections 2.9 and 2.10.

The Lie (super)algebras preserving symmetric non-degenerate bilinear forms B
We often denote the Gram matrix of the bilinear form B also by B, let aut(B) be the Lie (super)algebra preserving B. If B is odd and the superspace, on which it is defined, is of superdimension n|n, we write pe B (n) instead of aut(B).
Let p = 2 and g = pe B (n). The Lie superalgebra g consists of the supermatrices of the form Clearly, str X = 2 tr A. We also have g (1) = spe(n), i.e., spe(n) is of codimension 1; it is singled out by the condition str X = 0, which is equivalent to tr A = 0. The Lie superalgebra le(n; N |n) is, by definition, the Cartan prolong (id, pe(n)) * ,N . Over C, there is no shearing vector, and le(n) , generate le(n; N ). If N i < ∞ for at least one i, the additional part Irreg does not change while the regular part looks the same for any p > 2: In other words: there are vector fields corresponding to non-existing generating functions, like q and ξ 2 j . The prolong sle(n; N ) := (id, spe(n)) * ,N is singled out by the condition The operator ∆ is, therefore, the "Cartan prolong of the supertrace on g 0 " expressed as an operator acting on the space of generating functions. Modifications in the above description for p = 2. If p = 2, the analogs of symplectic (resp. periplectic) Lie (super)algebras accrue additional elements: if the matrix of the bilinear form B is Π 2n (resp. Π n|n ), then aut(B) consists of the (super)matrices of the form where B and C are symmetric, A ∈ gl(n).

(2.27)
Denote the general Lie (super)algebra preserving the form B as follows: Let ZD denote the space of symmetric matrices with zeros on their main diagonals.
The derived Lie (super)algebra aut (1) (B) consists of the (super)matrices of the form (2.27), where B, C ∈ ZD. In other words, these Lie (super)algebras resemble the orthogonal Lie algebras. On these Lie (super)algebras aut (1) (B) the following (super)trace (half-trace) is defined: The half-traceless Lie sub(super)algebra of aut (1) (B) is isomorphic to aut (2) (B). There is, however, an algebra aut(B), such that aut (1) (B) ⊂ aut(B) ⊂ aut(B), consisting of (super)matrices of the form (2.27), where B ∈ ZD, and any symmetric C (or isomorphic to it version of the Lie superalgebra with C ∈ ZD, and any symmetric B). We suggest that it be denoted as follows:

Central extensions
There is only one non-trivial central extension of spe(n) for p = 2, 3 existing only for n = 4. We denote it as because it was discovered by A. Sergeev (1970s, unpublished). For numerous non-trivial central extensions of versions of spe(n) and its simple subquotients for p = 2, 3, see [6]. Let us represent an arbitrary element A ∈ as as a pair The Lie superalgebra as can also be described in terms of the spinor representation. For this, we need several vectorial superalgebras. Consider po(0|6), the Lie superalgebra whose superspace is the Grassmann superalgebra Λ(ξ, η) generated by ξ = (ξ 1 , ξ 2 , ξ 3 ) and η = (η 1 , η 2 , η 3 ) with the Poisson bracket.
Clearly, if N = N ∞ , see (1.8), then le gen (n; N |n) consists of the following two parts, cf. equation (2.26): The part Irreg gen corresponds to the nonexisting generating functions ξ 2 i . Clearly, le gen (n; N |n) is contained in svect(n; N |n), and therefore le gen (n; N |n) = sle gen (n; N |n).
The difference between le gen (n; N |n) and le(n; N |n) is constituted by the space Irreg gen . The nonexisting generating functions ξ The correct p = 2 analogs of the complex Lie superalgebras sle(n) and spe(n) are, respectively, id, (pe(n)) (1) * ,N and pe(n) (1) . In [48], Lebedev considered g = pe(n), the derived algebras g (1) and g (2) , and the Cartan prolongs of these derived algebras playing the role of g 0 , whereas for g −1 he considered the tautological g 0 -module id. Clearly, g (1) consists of supermatrices of the form (2.27) with zerodiagonal matrices B and C, whereas g (2) is singled out of g (1) by the condition htr = 0. The corresponding Cartan prolongs only have the regular parts: Let a non-degenerate (anti)symmetric bilinear form B be defined on a superspace V ; let F(B) be the same form considered on F(V ), the same space with superstructure forgotten. Let h B (a; N |b) denote the Hamiltonian Lie superalgebra -the Cartan prolong of the ortho-orthogonal Lie superalgebra oo B (a|b) preserving the non-degenerate form B; its desuperization is h F(B) (a+b; N ), where N has no critical coordinates.
Remark 2.7. For N with N i < ∞ for all i and p = 2, the Lie superalgebra le (1) (n; N |n) is spanned by the elements f ∈ O(q; N |ξ), whereas each of the "virtual" generating functions q (2 N i ) i ∈ O(q; N |ξ) determines an outer derivation of le (1) (n; N |n).

Divergence-free subalgebras g of series h and le in the standard W-grading
These subalgebras are prolongations of subalgebras of 0th components of h and le consisting of traceless subalgebras; that is how these (super)algebras were described in [48].
It is possible, however, to describe various subalgebras of h 0 or le 0 , generated by (linear combinations of) quadratic monomials, by eliminating squares of indeterminates from the set of functions generating g 0 . In other words, constraints imposed on the shearing vector N corresponding to the space of generating functions determine various divergence-free subalgebras of h(n; N ) and le(n; N ).

spe a,b (n)
For p = 0, the meaning of spe a,b (n) is similar to that of svect a,b (0|n), but with d := diag(1 n , −1 n ). To define the analog of spe a,b (n) for p = 2, see line N = 7 in Table 25.2, observe that the codimension of spe(n) in m 0 , where m := m(n) is considered in its standard Z-grading, is equal to 2.
So, to pass from spe(n) to m 0 , we have to add two linearly independent elements, whereas to pass to spe a,b we have to add a linear combination of these elements with coefficients a and b. The question is: "can we single out these elements in a canonical way?" For p = 0. The identity operator (in matrix realization) is one of these elements. How to select the other element? There is no distinguished element in pe(n)\spe(n). But, if p = 0, there is an element diag(1 n , −1 n ) corresponding to a "most symmetric" generating function q i ξ i . For p > 2 this "most symmetric" element lies in spe(n) if p divides n and the choice of the linearly independent second element from pe(n) \ spe(n) becomes a matter of taste.
For p = 2, the situation becomes completely miserable. Now, the restriction of M q i ξ i to m −1 not only lies in spe(n) for n even, it coincides with the identity operator. So, in this case, there is no distinguished operator not lying in spe(n). What to do?
We suggest considering the elements of m 0 as operators acting not just on m −1 , but on the whole m − . If m 0 is thus understood, there are two well-defined linear forms and µ that single out spe(n) in m 0 :

On m and b
To pass from b(n; N |n) to m(n; N |n + 1), we have to add to b(n) 0 = pe(n) the central element; it will serve as a grading operator of the prolong. We see that m is the generalized Cartan prolong of (b(n) − , cb(n) 0 ). The commutant of m(n; N |n + 1) 0 is the like that of b(n) 0 = pe(n), so is of codimension 2. Hence there are two traces on m(n; N |n + 1) 0 , namely htr and , see (2.30), and therefore there are two divergences on m. One of them is given by the operator since this should be the mapping commuting (not super commuting) with m − , see [66]. The condition D τ (f ) = 0, i.e., just ∂ τ (f ) = 0 singles out precisely b(n).
The other divergence is given by the operator (2.32).

sb
The definition of sb(n; N ) is the same for any characteristic p (in terms of generating "functions" from an appropriate space F, see (1.9)): The direct analog of trace on m 0 is htr. On le, the prolong of htr is the operator ∆. But ∆ does not commute with the whole of m − . To obtain the m − -invariant prolong of this trace on m 0 , we have to express htr in terms of the operators commuting with m − (Y -type vectors in terms of [66]). Taking m − spanned by the elements we see that the operators commuting with m − are spanned by In terms of these operators, the vector field M f takes the form: and the invariant prolong of htr -the direct analog of divergence -takes the form: The condition ∆ m (f ) = 0 singles out the p = 2 analog of sm, whereas the condition we single out a subalgebra in the Lie algebra of contact vector fields which has no analogs for p = 2. Let us figure out how the parameter λ of the regrading po λ (2n; N ) := F(b λ (n; N )) depends on parameters a, b above; for a summary, see N = 6, 7 in Table 25.2. The space of b a,b (n; N ) consists of vector fields (2.31) whose generating functions satisfy equation (2.33); the regrading deg τ = deg q i = 1, deg ξ i = 0 for all i turns b a,b (n; N ) into the Lie superalgebra b λ (n; N ; n) whose 0th component is isomorphic to vect(0|n) and the (−1)st component is isomorphic to the vect(0|n)-module Vol λ of weighted λ-densities. Set To express λ in terms of the parameters a, b, we take an element in the 0th component of b a,b (n; N ) not lying in b(n; N ) and see how it acts on M 1 .
Proposition 2.8 (two exceptional deforms of the Poisson algebra). Desuperization of the simple Lie superalgebras introduced in (2.34) yields 2 exceptional serial Lie algebras that have no analogs for p = 2: In [58], we described the deformations of the Buttin algebra b(n) over C, having corrected a result due to Kochetkov [42] who described exceptional deformations of b λ (n) at certain values of λ; some of these deformations having an odd parameter. It is an open problem to obtain a version of [58] in the modular case.

On k and po for p = 2
Observe that po I (n; N ), a central extension of the Lie algebra h I (n; N ), is not a Lie algebra. Indeed, the bracket should be anti-symmetric, i.e., alternate, while {x i , x i } I = 1, not 0, so po I (n; N ) is a Leibniz algebra, not Lie algebra. Only h Π (2n; N ) has an analog of the familiar central extension; this nontrivial central extension is a correct direct analog of the complex Poisson Lie (super)algebra po(2n|0).
To pass from po(0|n) to k(1; N |2n), we have to add, as a direct summand, a central element to po(0|n) 0 = o (1) Π (n); it will act on the prolong of po(0|n) − , co (1) Π (n) as a grading operator. We see that the generalized Cartan prolong of po(0|n) − , co Thus, there are two traces on k 0 (1; N |2n), and hence there are two divergences on k(1; N |2n), like on m(n; L|n). These divergences are given by almost the same formulas as for m(n; L|n), where L = (N, 1) and 2 N is the hight of t, "almost" because ∂ t should replace ∂ τ .

2.11
Exceptional simple vectorial Lie superalgebras for p = 2 analogous to their namesakes over C We give detailed description of all exceptional simple vectorial Lie superalgebras over C and fields of characteristic 2 in the main text; for a summary, see Section 25. These Lie superalgebras constitute two non-intersecting sets as follows.
The complete Cartan prolong of its negative part: such is every Lie superalgebra of series vect, k and m in the standard grading, see (2.18) and each simple exceptional Lie superalgebra g of depth > 1, whose negative part in its W-grading is different from the negative part of the Lie superalgebras of series k or m in their respective standard gradings.
The complete Cartan prolong of its nonpositive part: such are the exceptional vectorial Lie superalgebras, and their desuperizations, see Tables 25.3 and 25.5, other than in the above paragraph; the corresponding gradings are explicitly given in Table (25.4).
The desuperizations of two nonisomorphic Lie superalgebras realized by vector fields on supervarieties of different superdimension might turn out to be vectorial Lie algebras realized on varieties of the same dimension. We distinguish these cases by indicating their depths as an index at the name (mb 2 (11; N ) and mb 3 (11; N ), and also kle 2 (20; N ) and kle 3 (20; N )); for the case of equal depths, we distinguish non-isomorphic algebras by a tilde: vle(9; N ) and vle(9; L), as well as kas(7; N ) and kas(7; L); for details, see respective sections.

A technical remark: natural generators of vectorial Lie superalgebras
This subsection is needed for calculations only. Let g = ⊕g i be a Weisfeiler grading of a given simple vectorial Lie superalgebra. We see that g −1 is an irreducible g 0 -module with highestweight vector H, and g 1 is the direct sum of indecomposable (sometimes, irreducible) g 0 -modules with lowest-weight vectors v i .
Over C, and over K for N = 1, the simple Lie superalgebra g is generated (bar a few exceptions) by the generators of g 0 , the vector H, and the v i . (For other values of N , we have to add the g 0 -lowest-weight vectors v k j ∈ g j for some j > 1 to the above generators; these cases are not considered.) So we have to describe the generators of g 0 , or rather of its quotient modulo its center.
If g 0 is of the form g(A) or its simple subquotient, we select its Chevalley generators, see [10]. If g 0 is an almost simple "lopsided", see Section 3.2.2 (in particular, of type pe, spe), but Z-graded Lie superalgebra, we apply the above-described procedure to g 0 : first, take its 0th components and its generators, then the highest and lowest-weight vectors in its components of degree ±1, etc.
If g 0 is semi-simple of the form s ⊗ Λ(r) vect(0|k), where s is almost simple, then we take the already described generators of vect(0|k) and apply the above procedures to s.
For a list of defining relations for many simple Lie superalgebras over C, and their relatives, see [27,29]. For defining relations for Lie algebras with Cartan matrix over K, see [5].

Introduction: overview of the scenery
In the Introduction (divided into two parts to ease digesting it) we give a brief sketch of the main constructions and ideas; for basic background, see Section 1. For further details, see [48,55,56]. All voluminous computations are performed with the help of the SuperLie package, see [26].
3.1 Goal: classification of simple finite-dimensional Lie algebras over K a.k.a. modular In 1960s, Kostrikin and Shafarevich suggested a method for producing simple finite-dimensional Lie algebras over K for any p > 0, together with the final list for p > 5. This list is explicit for simple Z-graded algebras; for the rest, it is somewhat implicit ("and deforms of Z-graded algebras"), see [40]. The above-mentioned deforms are often deforms of non-simple algebras the stock of which was not clearly described; this made this part of the KSh-method rather vague.

The original KSh-method
The initial ingredients are simple Lie algebras over C of two types: infinite-dimensional vectorial types (vect, svect, h, and k) with polynomial coefficients. The ingredient (3.1) yields via (3.3) one finite-dimensional Lie algebra; the ingredient (3.2) yields via (3.4) an infinite family of finite-dimensional Lie algebras over K depending on the shearing vector N . Each of the finite-dimensional Lie algebras thus obtained is either simple, or a "relative" of the simple Lie algebra over K (a central extension or a subalgebra in the algebra of derivations). Some of these simple Lie algebras can be deformed.
To describe the deforms is a rather complicated part of the KSh-method. (3.5) Let us clarify claim (3.5). Tables (3.8) and (3.11) show that some of simple Lie algebras are filtered deforms not of the simple Z-graded algebras, but of certain non-simple subalgebras of their Cartan prolongs (since their dimensions differ from those of simple algebras). The list of deforms was obtained in a roundabout way, avoiding computing the cohomology that describes a filtered deformation: 2) Skryabin [69,70] classified (for p > 2) all equivalence classes of symplectic forms (Skryabin called them Hamiltonian forms); some of Skryabin's difficult-to-obtain results hold for p = 2 as well.
Types of Lie algebras svect described by Tyurin and Wilson [74,76]. In the mid-1970s, Kac observed in [32] that the Lie algebra that preserves the volume element of the form h vol, where h ∈ O N i is invertible, can be a subalgebra of vect(m; N ) with finite coordinates of N . Let p > 2 and suppose that The results of Tyurin and Wilson, correct for p > 3, state that there are only the following three types of non-equivalent classes of volume forms, and hence filtered deforms with parameter ε ∈ K × of divergence-free algebras preserving them: where h is one of the following: Remarks 3.1.
1. For p = 3 and 2, these deformations of svect are also possible. For p = 3, nobody knows if there are other deforms, whereas for p = 2, there definitely is at least one more deform: its existence is the most spectacular result of this paper, see Section 14.
2. S. Tyurin described the Lie algebras of divergence-free type and got an extra type of volume form, as compared with Wilson's list (3.7), cf. [74].
3. S. Kirillov [38] verified Skryabin's remark in passing [71] for which i the ith derived algebra from Wilson's list (3.7) is simple, and found the dimensions of these simple Lie algebras: Hamiltonian Lie algebras h described and classified by S. Skryabin [69,70]. Let h(2k; N ) or h ω 0 (2k; N ) be the Z-graded Lie algebra preserving the symplectic form The only non-isomorphic filtered deforms of h ω 0 (2k; N ) with parameter ε ∈ K × are h ω i (2k; N ), where i = 1, 2, which preserve the following respective forms (of type 1 and 2 in Skryabin's terminology): and where the non-equivalent normal shapes of the indecomposable matrices A = (A i,j ) can only be equal for p > 2 to one of the following: where J k (λ) is a Jordan k × k block with eigenvalue λ, and J k,r (λ) is a k × k block matrix with blocks of size r × r, so k = r × n for some r, n ≥ 1: The two conditions on J k,r (λ) and C k . 1) The case with J k,r (λ) occurs only when and, furthermore, N ir−j = N ir for all i = 1, . . . , 2n and all j = 1, . . . , r − 1; i.e., r indeterminates in each of the 2n successive groups have equal heights.
The case with C k occurs only when condition (3.10) is violated.
2) Let G be the group generated by the cyclic permutations of the row vectors of length k. Then, the identity element is the only permutation in G that simultaneously fixes the two vectors a = (N 1 , . . . , N k ) and b = (N k+1 , . . . , N 2k ).
It suffices to consider representatives of equivalence classes of pairs (a, b) under the G-action. 1. Over C the supervarieties of parameters of deformations of Poisson and Hamiltonian Lie superalgebras can differ, see [58]. For p = 2, there is at least one new type of deform: a 1parametric family of non-isomorphic deforms different from the above -desuperisations of b a,b (n; N ).
2. S. Kirillov [38] determined the i for which the ith derived algebra of the Hamiltonian Lie algebra from Skryabin's list [69] is simple and what its dimension is equal to: (3.11)

True and semi-trivial deforms
In particular, the amount of infinitesimal deformations is overwhelming and even frightening as p becomes small (p = 3 or -a horrible case -p = 2). We recall reasons not to be too frightened; besides, the KSh-method had been considerably improved over the past years.
The abundance of deforms of simple Lie (super)algebras for p > 0, especially overwhelming for p = 2, is somewhat misleading. It is occasioned by semi-trivial deforms each of which is given by a cocycle representing a nontrivial cohomology class but, though integrable, yields a deform isomorphic to the initial algebra. For a description of many semi-trivial deforms, see [13]. We say that a nontrivial and nonsemi-trivial deform is a true deform.
The Lie (super)algebra g is said to be rigid if it has no true deforms; until recently we thought that semi-trivial deforms existed only if p > 0, but a more careful study of the literature shows they are a universal phenomenon [61].
If p = 3, we conjecture the classification: the examples obtained by Cartan prolongation (see Section 1.6) of appropriate parts of Lie algebras with Cartan matrix [7,28], exhaust the list of "standard" examples some of which were discovered by Frank, Ermolaev and, mainly, Skryabin. For an (incomplete at the moment) list of true deforms of several "standard" algebras, see [8,45,46,47,70], and [14] in which an earlier claim concerning deforms is corrected.
If p = 2, we are still completing the stock of "standard" examples.

Amendments to the formulation of the goal
On several occasions P. Deligne told us what we understood as follows (for Deligne's own words, and several open problems, see [49]): "In positive characteristic, the problem "classify ALL simple Lie (super)algebras, and their representations" is, perhaps, not very reasonable, and definitely very tough; investigate first the restricted case related to geometry, and hence meaningful." Following Deligne's advice, we investigated several plausible notions of restrictedness for p = 2 in [12] and gave explicit expressions of the restriction maps for several types of simple Lie algebras and superalgebras in [11]. Nevertheless, even to describe restricted Lie (super)algebras one often needs nonrestricted ones; for more serious examples of their usage, see [40].
In this paper, we concentrate on simple Lie (super)algebras, keeping in mind that algebras of the following types are no less important than simple ones: Lie (super)algebras of the form g(A) where A is indecomposable, see [7,10].
Central extensions and algebras of derivations of the known simple Lie (super)algebras (see Section 14.1 and [6]). The algebras of these two types will be called relatives of the corresponding simple Lie (super)algebras and each other.
The generalized Cartan prolongs (g − , g 0 ) * ,N with g 0 close to simple, see Sections 1.6 and 14.1.
Restricted closures of nonrestricted simple Lie algebras.

"Standard" modular Lie algebras
Dzhumadildaev and Kostrikin [41] suggested simplifying the KSh-method by skipping the step over C and considering certain "standard" modular Lie algebras from the very beginning, further deforming them and their "relatives". On the other hand, the stock of "standard" examples should include, if p < 7, certain non-simple Lie algebras, see [25,41,72]. The snag is: we have no idea how to select them.
Until the year 2000 or so, it was believed that the initial KSh-method produces all simple Lie algebras only if p > 5. This belief was based on insufficient study of deformations and too narrow a choice of "standard" examples: as shown in [41], the Melikyan algebra, indigenous for p = 5, are deforms of Poisson Lie algebra which should be considered "standard" and processed via the KSh-scheme (3.1)- (3.5).
What examples should qualify as "standard"? In [51], the improvement of the KShmethod suggested in [41] was developed further by eliminating the vectorial simple Lie algebras from the input of the KSh-method thus diminishing the stock of "standard" simple Lie algebras. In the new procedure, the role of generalized Cartan prolongation (complete or partial), see Section 1.6 and especially Section 1.6.1, becomes even more important than in the KSh-procedure. This approach definitely works for p > 3, and conjecturally works for p = 3.

Splitting the problem into smaller chunks
All simple Lie algebras are of the following two types: the root system of a "symmetric" algebra contains the root −σ of the same multiplicity as that of σ for any root σ; the algebras with root systems without this property are said to be "lopsided ". This paper is devoted to the study of lopsided algebras, but "symmetric" Lie (super)algebra will be needed in the process Symmetric algebras. A significant quantity of symmetric simple Lie algebras consists of algebras g(A) with indecomposable Cartan matrix A or simple "relatives" of such algebras of the form g (i) (A)/c, where 4 g (i) (A) is the ith derived algebra of g(A) and c is the center of g (i) (A).
For any p, finite-dimensional Lie algebras g(A) with indecomposable Cartan matrix A, and their simple relatives, were classified in [75] with an omission; for corrections, see [34,71], where no claim was made that these were the only corrections needed; for this claim with a proof, a classification of Lie superalgebras of the form g(A) with indecomposable Cartan matrix A, and their simple relatives, and precise definitions of related notions, see [10].
Lopsided algebras: the set they constitute is a virtually virgin territory a part of whichvectorial Lie (super)algebras -we investigate through the whole of this text.

Cartan prolongations of Lie (super)algebras with Cartan matrix
It turns out that every known Z-graded simple Lie algebra for p > 2 is obtained as a (generalized, perhaps partial) Cartan prolong of the non-positive part of a Lie algebra g(A) with an indecomposable Cartan matrix A. For p > 3 this follows from the classification.

Super goal
Although Lie super algebras appeared in topology in the 1940s (over finite fields, often over Z/2), the understanding of their importance dawned only in the 1970s, thanks to their applications in physics. This understanding put the problem "classify simple Lie superalgebras" on the agenda of researchers. Over C, the finite-dimensional simple Lie superalgebras were classified by several teams of researchers, see reviews [33,37]. The classification of certain types of simple vectorial Lie superalgebras was explicitly announced in [33], together with a conjecture listing all primitive vectorial Lie superalgebras; for the first counterexamples, see [1,50].
A classification of the simple vectorial Lie superalgebras over C was implicitly announced when the first exceptional examples were given [63,64,65] and explicitly at a conference in honor of Buchsbaum [57]. The claim of [33] was corrected in [55] (the correction contained both the complete list of simple vectorial algebras, bar one exception later described in [64], and the method of classification of simple Z-graded Lie superalgebras of depth 1) and in a series of papers [17,19,20,21,35,36], where the proof in the case of Z-grading compatible with parity was given; for further corrections and proofs, see Section 13. The classification is not completed till today: there is no classification of deformations with odd parameters.
Although to complete the classification of the simple finite-dimensional Lie superalgebras over K for p "sufficiently big" (say > 7) will be a more cumbersome and excruciating task than that for Lie algebras, the answer (conjectural, but doubtless) is obvious: to get restricted superalgebras, take the obvious modular analogs of the complex simple Lie superalgebras (of both finite-dimensional and of infinite-dimensional vectorial considered for the shearing vector N = 1, see definition (1.8)) passing to the derived algebra and quotients modulo center if needed; to get nonrestricted superalgebras, consider true deforms, see Section 3.1.2, of the abovementioned analogs (for N unconstrained, speaking about vectorial algebras). For p "small", the classification problem becomes more and more involved, see, e.g., [9,30]. Nevertheless, in the two cases the classification is obtained: For any p, the super goal is reached for Lie superalgebras of the form g(A) with indecomposable Cartan matrices A or its "relative", see [10]. Either g = g(A) or its "relative" of the form g (i) /c, where g (i) is the ith derived algebra of g and c its center, is simple. For deforms, see [8].
Amazingly, the super goal is reached if p = 2, see [12], with a catch: modulo the classification of simple Lie algebras, i.e., without an explicit list of all examples. 5 Here, we contribute to a conjectural list of "standard" simple Lie algebras (conjecturally a tame problem); in particular, we explicitly describe simple vectorial Lie algebras analogous to those over C.
p > 5. The conjecture [51] is easy to formulate ("take direct characteristic-p versions of the simple complex Lie superalgebras of the form g(A), queer, and vectorial with polynomial coefficients, and their deformations"), but to describe deformations of even the symmetric ones (of the form g(A) and queer) is not easy (for partial results, see [8]). p = 5. Most plausible conjecture is like for p > 5. Observe that there are indigenous p = 5 examples of the form g(A), p = 3. We have discovered several new vectorial Lie superalgebras, see [8,15], and of the form g(A), see [10].

Getting simple Lie algebras from simple Lie superalgebras if p = 2
If p = 2, there are two methods for constructing a simple Lie superalgebra from a simple Lie algebra, and every simple Lie superalgebra is obtained by one of these two methods; for an amazingly short proof, see [12]. Reversing the process we recover a simple Lie algebra given any simple Lie superalgebra.
Even before these two methods were known, it was clear that one can get a Lie algebra from any Lie superalgebra as follows. Observe that for any odd element x ∈ g in any Lie superalgebra g over any field K, we have [x, x] := 2x 2 ∈ U (g). That is why if p = 2, then one needs a squaring x −→ x 2 for any odd x ∈ g; together with the brackets of even elements with all other elements, it is the squaring that defines the multiplication in any Lie superalgebra (for details, see Section 1.2.3), while the bracket of odd elements is the polarization of the squaring. Hence, For p = 2, every Lie superalgebra with the bracket as multiplicationwe forget the squaring -is a Z/2-graded Lie algebra. (3.12) To classify simple Lie superalgebras is a much more difficult task than to classify simple Lie algebras of the same type: the former is based on the latter as well as on careful study of the representation theory of Lie algebras. In [39], it was, nevertheless, suggested -for p = 2 -to reverse the process: Let F be the desuperization functor forgetting squaring, see (3.12). To obtain simple Lie algebras for p = 2, (A) apply the functor F to every simple Lie superalgebra g; (B) single out the simple Lie subalgebra s(F(g)) of F(g). Clearly, s(F(g)) is uniquely recoverable by inverting one of the two superization processes (either queerification or "method 2") that had lead to g, see [12].
Observe immediately that the idea of [39] just to apply F to the simple Lie superalgebra g to get a simple Lie algebra, see (3.12), was naive and partly wrong: the example of psq(n) should have hinted at importance of item (B) in the process (3.13). Understanding of this subtlety came together with the description of the two methods of superization of any simple Lie algebra as the only means to obtain any simple Lie superalgebra, see [12]. For the simple vectorial Lie is not yet handable". This is no wonder: although Skryabin classified symplectic forms in 1985, the answer was published only in 1991, see [70], three years after [4] appeared (and the details of [70], obtained in 1985, became available only recently, see [69]). The explicit formula for the bracket in any of the deformed Lie algebra of Hamiltonian vector fields is not published to this day and can be found only in Kirillov's Ph.D. Thesis only (in Russian), not in its published summary [38].
superalgebras, in particular, exceptional ones, just by forgetting squaring we get a simple Lie algebra. Let F −1 = s(F(−)) denote the complete desuperization− the composition of F and application of item (B) of (3.13).
Two reasons to take the direction of study opposite to a seemingly reasonable one: (a) Although the classification of the simple vectorial Lie superalgebras over C was only conjectured at the time [39] was written, the list of known examples was already wider than that of known simple vectorial Lie algebras for p = 2, and was (and is, as we demonstrate in this paper) able to provide new simple examples.
(b) The results of [25,72] show that a "frontal attack" on the classification for p = 2 is likely to be much more excruciating than that performed for p > 3 by Premet and Strade. Even to classify restricted Lie algebras for p = 2 will be much more difficult problem than that Block and Wilson solved for p > 5, see [4]. (Even if we confine ourselves to the classical definition of restrictedness, while certain examples, which should be considered as "classical", have another version of restrictedness, see [12].) So, a plausibly complete inventory of simple examples will be helpful. Our interpretations of the Lie (super)algebras are of independent interest.
Here, after a long break, we continue exploring method (3.13). It provides us with new examples of simple vectorial Lie algebras of the form F(g), where g is a modular, indigenous for p = 2, version of a simple vectorial Lie superalgebra over C. The two methods of superization (see [12]) applied to F(g) bring many more simple Lie superalgebras than g, most of them new.

Forgetting the superstructure if p = 2
Applying F to the serial vectorial Lie superalgebra g(m; N |n) we get the Lie algebra F(g) m + n; N , see Table 25.2; these Lie algebras are not necessarily simple, but their simple derived algebras are; here N = (N , 1, . . . , 1) with the last n coordinates equal to 1.

Parameters of the Lie superalgebra that change under desuperization
Here are several examples: The unconstrained shearing vector N u , see Section 1.9, of the vectorial Lie algebra F(g) may depend on more parameters than the shearing vector N u of g. for F(vect(a; N |b)) = vect a + b; N . In all cases, except for vle 4; M |3 , the tilde over any shearing vector L is understood in the above sense: it enlarges the set of coordinates of L acquiring the coordinates of the desuperized odd indeterminates.
The same applies to the desuperizations of the Lie superalgebras of the series k, h, m, le and their divergence-free subalgebras.
The abstract Lie superalgebra g realized as vectorial Lie superalgebra, g(a; N |b), depending on a even and b odd indeterminates, can be realized in several ways as g(a; N |b; r) by means of Weisfeiler filtrations or associated regradings r, see Section 2.4. This g(a; N |b; r) can be interpreted as the (generalized) Cartan prolong of the nonpositive part of g in the corresponding grading, see Section 1.6.1.
The Lie algebra obtained by desuperization might acquire new properties which its namesakes for p = 2 do not have. For example, the Lie algebra po(2n; N ) = F(b λ (n; n; N )) has a deformation depending on a parameter λ ∈ KP 1 ; the corresponding non-isomorphic (for different values of λ) deforms are additional to the well-known one, which for p = 0 is called the result of the quantization.
Desuperization F might turn distinct (types of) Lie superalgebras into one (type of) Lie algebras: -the Lie superalgebras of types k and m in the standard grading (2.18) turn into k; -the Lie superalgebras of types h Π and le in the standard grading (2.18) turn into h Π .

Comment on Volichenko algebras in characteristic 2
The notion of Volichenko algebra 6 , which is an inhomogeneous (relative to parity) subspace h ⊂ g of a Lie superalgebra g closed with respect to the superbracket in g, was introduced in [39]. For a classification of simple (without any nontrivial ideals, both one-sided and twosided) finite-dimensional Volichenko algebras over C under a certain (hopefully, inessential) technical assumption, and examples of certain infinite-dimensional algebras, see [53,54].
The results of [31] suggested that we look at the definition of the Lie superalgebra for p = 2, see Section 1.2.3, more carefully. If one does this, it is not difficult to deduce that if p = 2, the Volichenko algebras are, actually, Lie algebras. (3.14) In [31,39], the fact (3.14) had not been understood, and therefore there is no need to consider these papers or Volichenko algebras in characteristic 2 while searching for simple Lie algebras. Unlike desuperizations of Lie superalgebras, which are worth investigating.

Generalized Cartan prolongation
This is a principal procedure for getting vectorial Lie (super)algebras over C, the following fact is well-known [77]: given a simple Lie algebra of the form g(A), and its Z-grading, the generalized prolong of the nonpositive (with respect to that grading) part of g(A) is isomorphic to g(A), bar two series of exceptions corresponding to certain simplest gradings of the embedded algebras -sl(n + 1) ⊂ vect(n) and sp(2n + 2) ⊂ k(2n + 1) -and the ambients are the exceptional prolongs. In [51], it is shown how to obtain simple Lie algebras over C of the two types of prime importance for the classification procedure over K: finite-dimensional and vectorial. Namely, by induction and using (generalized, in particular, partial) Cartan prolongation, see Section 1.6. First, one thus obtains all finite-dimensional simple Lie algebras (each of them has a Cartan matrix); during the next step one obtains all four series of simple vectorial Lie algebras, by considering not only complete Cartan prolongs as in equation (4.1), but also partial ones.
The same method works to obtain Z-graded simple Lie algebras for p sufficiently big and with new standard examples added. After that, there still remain considerable technical problems: namely, to describe the deforms and to classify non-isomorphic deforms.
For any characteristic, the super version of classification of simple Lie algebras is much more complicated than its non-super counterpart: we have to supply the input with several more types of algebras, but the main procedures are still the same: generalized, especially partial, prolongations and deformations.
In several papers (e.g., [8,7,15]), we have considered simple Lie (super)algebras, and have investigated the prolongs of the nonpositive parts relative to their Z-gradings with one (or two if p = 2) pair(s) of Chevalley generators being of degree ±1, and the other generators being of degree 0.
Here we consider serial and exceptional simple vectorial Lie superalgebras over C and desuperizations of their analogs for p = 2. Realization of a given Lie (super)algebra g in terms of vector fields implies that g is endowed with a filtration; one of these filtrations, called the Weisfeiler filtration, is the most important, see equation (2.17). Associated with the filtration is the grading; for brevity, the Weisfeiler filtrations and gradings are referred to, respectively, as W -filtration and W-grading. For several vectorial Lie algebras over fields of characteristic 2, we investigate the following problem answered by the fact (4.1) over C: when (g − , g 0 ) * ,N u g and when the prolong strictly contains g?
We consider only the Z-gradings of the finite-dimensional vectorial algebras corresponding to the W-gradings of their infinite-dimensional versions corresponding to N u . For examples of Lie (super)algebras g that differ from the prolong of the nonpositive part of a regrading of g, see [7].

"Hidden supersymmetries" of Lie algebras
It is sometimes possible to endow the space of a given simple Lie algebra g with (several) Lie superalgebra structures. For example, for g = sl(n) over any ground field, consider any distribution of parities (of the pairs corresponding to positive and negative simple roots) of the Chevalley generators; we thus get Lie superalgebras sl(a|b) for a + b = n in various supermatrix formats. The sets of defining relations between the Chevalley generators corresponding to different formats are different. It is, clearly, possible to perform such changes of parities of (pairs of) Chevalley generators for any simple Lie algebra but, except for sl(n), the simple Lie superalgebras obtained by factorization modulo the ideal of relations [10] are infinite-dimensional unless p = 2.
If p = 2, the following fact is obvious (here x −→ x [2] is the restriction mapping and x −→ x 2 is the squaring): 7 any classically restricted and Z/2-graded Lie algebra g = g + ⊕ g − can be turned into a Lie superalgebra by setting x 2 := x [2] for any x ∈ g − . (4.2) In [12], a (rather unexpected) generalization of the possibility (4.2) is described: every simple Lie algebra g can be turned into a simple Lie superalgebra by slightly enlarging its space if g is not restricted. This generalization, and a "queerification", are the two methods which, from every simple Lie algebra, produce simple Lie superalgebras, and every simple Lie superalgebra can be obtained in one of these two ways, as proved in [12]. These two methods applied to the simple Lie algebras we describe in this paper yield a huge quantity of new simple Lie superalgebras, both serial and exceptional. We do not list them; this is routine, modulo the far from routine, as shown in [43], job of describing all Z/2-gradings of our newly found simple Lie algebras.

From C to K
We consider the W-grading (of the desuperized p = 2 version of the simple vectorial Lie superalgebra over C) for which the nonpositive part, especially the 0th component, is most clear. Then, we consider the regradings described in Table (25.4), and perform generalized Cartan prolongation of the nonpositive (or negative) part of the regraded algebra in the hope of getting a new simple Lie (super)algebra, as in (4.1). When this approach is inapplicable since there is no visible analog, suitable for p = 2, of the Lie (super)algebras we worked with over C, we consider the description of the Lie superalgebras as the sum of its even and odd parts, desuperize this description and only after this consider the W-grading of the desuperization.

Our main results
Observe that every modular analog of one vectorial Lie (super)algebra over C is usually a family of algebras depending on N ; by abuse of speech we often skip the word "family" (of algebras) and talk about one algebra having in mind the extra parameter N .
Of the simple Lie superalgebras that can be obtained by the two methods described in [12] from the simple Lie algebras we describe here all but one (the initial one, the one we desuperized) are new. In more details the Lie algebras obtained by desuperization are described in the following Theorems 4.1-4.5 summarizing respective sections with proofs.    (ii) For p = 2, all their finite-dimensional analogs, and their desuperizations, remain regradings of each other, with one exception: the 3 indeterminates of kas 7; K are defined to be constrained, Par K u = 3.
One -for p = 2 -exceptional family (kas) yields -for p = 2 -two families:  Other results. We explicitly describe characteristic-2 Shen's version (see [68] and Section 23 here) of both g(2) and the Melikyan algebra as vectorial Lie algebras.
We single out the divergence-free subalgebra sh Π 2n; N of the Lie algebra of Hamiltonian vector fields h Π (2n; M ) = F(le(n; N |n)), discovered in [48], by imposing constraints on M .

Open problems
1. In this paper, for serial simple vectorial Lie superalgebras, we considered 2 W-gradings of F sb(2 n − 1) (the standard and of depth 1). We did not consider 32 W-gradings of the remaining simple vectorial Lie superalgebras, see Section 1.16 in [56]. The fact (4.1) suggests that we should investigate if these W-gradings yield new simple Lie algebra.
3. Are the partial prolongs described in Section 19.1 and in equation ( 5 The Lie superalgebra vle(4|3) over C and its p > 2 versions 5.1 Recapitulations, see [67] In the realization of le(3) by means of generating functions, we identify the space of le (3) with Π(C[θ, q]/C · 1), where before the functor Π is applied θ = (θ 1 , θ 2 , θ 3 ) are odd and q = (q 1 , q 2 , q 3 ) are even, see (2.7). In the standard grading deg Lie of le (3), we assume that deg q i = deg θ i = 1 for all i, and that the grading is given by the formula The nonstandard grading deg Lie;3 of g = le(3; 3) is determined by the formulas deg 3 θ i = 0 and deg 3 q i = 1 for i = 1, 2, 3, This grading of g = le(3; 3) is of depth 1, and its homogeneous components are of the form: In particular, g 0 vect(0|3), and g 1 is an irreducible g 0 -module with the lowest-weight vector q 2 1 . The whole Lie superalgebra le(3; 3) is the Cartan prolong of its nonpositive part and the component g 1 generates the whole positive part.
To obtain vle(4|3), we add the central element d to the 0th component of le(3; 3); so that ad d is the grading operator on the Cartan prolong of its nonpositive part; this prolong is strictly bigger than le(3; 3) C · d.

Introducing indeterminate y (as well as x i and ξ i )
Under the identification where f, g ∈ C[x, ξ]; the operators B g and ∆ are given by the formulas There are two embeddings of le(3) into vle(4|3). The embedding i 1 : le(3) −→ vle(4|3) corresponds to the grading le(3; 3). Let us reproduce the explicit formulas from [67]: Let us clarify notation of indeterminates. At the beginning, the indeterminates describing le in any grading are denoted by q and θ, while the indeterminates describing vle are denoted by x and ξ. Hence, introduce the passage i from one set of indeterminates to the other, andf : where y is treated as a parameter and (i, j, k) ∈ A 3 (even permutations of {1, 2, 3}).
In what follows we will investigate the case of shearing vectors with finite coordinates.
Proof . For p > 2, the component g 1 also retains its structure as Direct observation gives the answer: i.e., we should add "virtual" (non-existing for the given M ) elements f = x (s i ) i for i = 1, 2, 3. Since due to (5.5) To obtain D = h s ∂ y , we should add to the space of generating functions any of the "virtual" functions for i = 1, 2, 3, and j, k = i, j = k.
Modulo the kernel (5.8) of the map D 0,− only one "extra" generator suffices; for definiteness, we select g s := x Since the Lie superalgebra vle(4; M |3) has a grading operator, it follows that D 0,gs ∈ vle (1) 4; M |3 for p > 2. Moreover, as was shown in [67] for p = 0 (but the formulas remain true for any p > 0), If p > 2, the fact (5.4) does not hold. We have If M i > 1 for i = 1, 2, 3, then sdim g 1 = 28|27 remains the same as over C and any other p > 2. In this and two next sections we prove Theorem 5.2 for the case p = 2.
There are three W-gradings of vle with unconstrained shearing vector. In this and two next sections, we consider each of these gradings for p = 2, describe the corresponding 0th and 1st components of the regraded Lie superalgebra vle, and calculate partial prolongs. Next, we consider desuperizations of each of them. As a result, we get a new simple Lie superalgebra and a new simple Lie algebra in p = 2. Unfortunately (we'd like to get new simple algebras!), there are no partial prolongs.
The Lie superalgebra vle 4; M |3 for p = 2 is a direct reduction modulo 2 of the integer form, with divided powers as coefficients, of the complex vectorial Lie superalgebra vle(4|3).
First of all, let us define squares of odd elements for the Lie superalgebra le(n; M ), cf. equation (2.14): and have in mind that if p = 2 and M i < ∞ for all i, the Lie superalgebra g = le(n; M ) is not simple: the generating function of the maximal degree q (s 1 −1) 1 q (s 2 −1) 2 · · · q (sn−1) n θ 1 θ 2 · · · θ n does not belong to g (1) , the latter being simple.
For p > 2, we just reduce the expression (5.1) modulo p. For p = 2, we cannot just reduce the expression (5.1) modulo 2; we should modify it. Indeed, the system of equations on the coefficients of the field D ∈ vle(4|3) whose solution is given by the formula (5.1) contain coefficients 1 2 , see [67]. The vector field D ∈ vle 4; M |3 is of the form (recall formula (5.2) for B g ): (multiplied the equations by 2) and solved them in the same way as it was done in [67].
For p = 2, unlike the case p = 2, this Lie superalgebra g = vle 4; M |3 is not simple, but g (1) is simple, its codimension in g is equal to 2: for f = q For p = 2, the structure of the g 0 -module g 1 differs drastically from that for p = 2. Instead of two lowest-weight vectors, we have three of them. Besides, these three lowest-weight vectors do not describe the whole complexity of the module.
The submodule i 1 (le(3; 3) 1 ) has a complicated structure. To describe it, observe that for any vectorial Lie (super)algebra expressed in terms of generating functions, the shearing vector can be considered either (1) on the level of generating functions (let us denote it M in this case) or (2) on the level of coefficients of vector fields they generate (let us denote it M in this case).
Accordingly, for M unconstrained, the component le 3; M ; 3 1 is of the form: This component contains submodules corresponding to the minimal values M i = 1 for some i. To describe g 1 as g 0 -module, consider the submodule W 0 := le (1) (3; 3) 1 = Span(q i q j ϕ(θ) for any i, j and any function ϕ).

It is irreducible. It is glued to the submodules
in each of which W 0 is a submodule, but not a direct summand. Each W i can be further enlarged to the module W i,θ := W i Span q Let us describe the subalgebras of vle 4; 1|3 , the partial prolongs. In what follows we will often use the following

Notational convention: on partial prolongs
Let v i be a lowest-weight vector of the g 0 -module g 1 and V i the submodule generated by v i . Let g k,(i) be the kth prolong "in the direction of V i ⊂ g 1 ", i.e., kth prolong of (g − , g 0 , V i ). (6.2) Consider the g 0 -submodules W ⊂ g 1 not contained in i 1 (le (3; 3)). There is only one such submodule V 3 ⊂ W generated by v 3 , see (6.3). The g 0 -module g 1,M has the following three lowest-weight vectors expressed in the form D f, g , and also as i 1 (−) or i 2 (−): By increasing the value of some of the coordinates M i we enlarge g 1,(3) = vle 4; 1|3 1 . As g 0 -module, g 1,(3) is of the following form: where g 1,(3) /(W 1 + W 2 + W 3 ) (g −1 ) * , and (W 1 + W 2 + W 3 )/W 0 is the trivial 0|3-dimensional module, and where sdim W 0 = 12|12.

Convention: on partial prolongs "of no interest"
In what follows, we do not investigate partial prolongs with [g −1 , g 1,(i) ], see (6.2), smaller than g 0 if the [g −1 , g 1,(1) ]-module g −1 is not irreducible: no such prolong can be a simple Lie (super)algebra with the given nonpositive part.

Desuperization
We have g 0 c(vect(3; 1)), and g −1 F/K. The dimensions of the positive components of vle(7; 1) and its simple derived algebra (in parentheses) are given in the following table; so dim vle (1) 7; 1 = 94: g1 g2 (or g .  The Lie superalgebra vle(3; N |6) := vle(4; M |3; K) is the complete prolong of its negative part, see Section 2.11. A realization of the weight basis of the nonpositive components by vector fields is as follows, where the w i is a shorthand notation for convenience: gi the basis elements The g 0 -module g 1 has the following lowest-weight vectors:

No simple partial prolongs
For N unconstrained, dim g 1 = 18. The module V 1 generated by v 1 is 6-dimensional, and the module V 2 generated by v 2 is 8-dimensional; V 1 ⊂ V 2 . Critical coordinates of the unconstrained shearing vector : N 4 , . . . , N 9 .
8 A description of vle 9; N := F(vle(5; N |4)) for p = 2 Whenever possible in this section, we do not indicate the shearing vectors. This Lie superalgebra is the complete prolong of its negative part, see Section 2.11. For p = 0, the g 0 -action on g −1 is that on the tensor product of a 2-dimensional space on the space of semi-densities in 2 odd indeterminates. So it is not possible to just reduce modulo 2 the formulas derived for the characteristic 0. We have to understand how g 0 acts on g −1 when p = 2. For this, we use the explicit form of elements of vle(4|3) for p = 2, see equation (6.1).
Note that the mapping Le f −→ D (f,0) determines a Lie superalgebra isomorphism between le(3) and its image in vle. However, first, the mapping D 0,− : g −→ D (0, g) has the kernel: Taking equation (5.2) into account, these conditions are equivalent to following conditions: The grading of the Lie superalgebra we are interested in is induced by the following grading of the space of generating functions: Therefore (here we introduce the 9 indeterminates z 1 , z 2 , z 3 , z 8 , z 9 (even) and z 4 , z 5 , z 6 , z 7 (odd) of the ambient Lie superalgebra vect(5; N |4) containing our g) and ∂ i := ∂ z i gi the basis elements in terms of vect(5; N |4) because, for the nonzero vector fields of the form D (0,g) lying in g − , we have, thanks to conditions (8.2), the following identifications: Because the tautological representation of sl (2) is isomorphic to its dual, we identify using the rules listed in Table (8.4). The table also contains the explicit form of the vector fields D (f,0) ∈ g −1 needed to calculate the action of the fields of the form D (0,g) ∈ g 0 on g −1 (the action of the fields of the form D (f,0) ∈ g 0 can be computed in terms of generating functions and the bracket in le).
Proof . Let us begin with fields of the form D (f,0) . As we have already noted above, [D (f 1 ,0) , D (f 2 ,0) ] = D ({f 1 ,f 2 },0) , and hence the action of such fields can be described in terms of the generating functions and the Buttin bracket.
If f = i,j=2,3 a ij x i ξ j and ∆(f ) = 0, then f acts on g −1 as thanks to our identification, corresponds to the action of the operator ( a 22 a 23 a 32 a 33 ) ⊗ 1; i.e., the elements of this form span the subspace sl(V ) ⊗ 1 ∈ End(W ).
Analogously, the functions of the form f = ξ 1 i,j=2,3 a ij x i ξ j such that ∆(f ) = 0 act on g −1 as 2 . In terms of generating functions, f acts as Le f , i.e., as the vector field (operator) X = x 2 ∂ ξ 2 . Looking at Table (8.4) we deduce that this X acts as non-zero only on ξ 2 and ξ 1 ξ 2 . Explicitly, X(ξ 2 ) = x 2 , and X(ξ 1 ξ 2 ) = ξ 1 x 2 . This is precisely what is written in the first row of Table (8.6).

Desuperization
For N unconstrained, the critical coordinates are those that correspond to the formerly odd indeterminates. 9 kle 15; N := F(kle(5; N |10)) Whenever possible in this section, we do not indicate the shearing vectors. The Lie superalgebra kle(5; N |10) is the complete prolong of its negative part, see Section 2.11.
To determine the component g 0 of g, we have to consider a linear combination of two elements: the central element Z commuting with the image of svect(0|4) in osp(6|8) and an outer derivation, say D = ξ 1 ∂ ξ 1 ∈ vect(0|4).
To realize the Lie superalgebra g by vector fields, we use the representation of the even part of g as svect(5; M ) and its odd part as Π dΩ 1 (5; M ) : whatever the Z-grading of g, the components g0 and g1 have the needed nonpositive part. For convenience, we use gl(5)weights of the elements of g, having added the outer derivation -the grading operator -to svect(5; M ).
Let u 1 , . . . , u 5 be a basis of the space U we used to define svect(U ) and dΩ 1 (U ). In our grading, deg(u 5 ) = 2 and deg(u i ) = 1 for i < 5. Let x 1 , . . . , x 15 be the desuperized indeterminates. Then, The functor Π is interpreted as multiplication (tensoring) by the 1-dimensional module whose generator Π has the following weight w to make the weight and degree compatible: To get rid of fractions, we multiply all weights by 2; assuming that deg du i = deg u i we have Now, the weights are symmetric in the sense that if there is an element of weight (2, 0, 0, 0, 0), there should be elements whose weight have all coordinates but one equal to 0, one coordinate being equal to 2. This symmetry helps to find correct expressions of the vector fields in each component. Thus, the weights of the indeterminates in the new grading are as follows: The degree is equal to one half of (the sum of the first 4 coordinates plus the doubled fifth one). In equation (10.2) we give the basis of the negative part and generators of the 0th component. It is possible to generate the semi-simple part of g 0 by just 1 positive and 4 negative generators, or 4 positive and 1 negative ones, but for symmetry we give 4 and 4 of them. These 8 generators do not generate the element D + Z ∈ [g 1 , g −1 ] of weight (0, 0, 0, 0, 0), so we give it separately.
Let us express the basis of g −1 in terms of the u i introduced in (10.1): , The Lie superalgebra kle(11; N |9) is the prolong of the negative part. For a basis of the negative part we take the following elements, see (12.2). For their weights we take We select the degree of Π so as to ensure the correct degrees of the ∂ x i , see (12.2), where by abuse of notation ∂ i := ∂ x i . Looking at the expression of ∂ {0, 0, 0, 0, −2} → ∂2 + x12∂17 + x13∂18 + x14∂19 + x16∂20, No simple partial prolongs. The critical coordinates of the shearing vector for kle 2 20; N are those corresponding to the formerly odd indeterminates. Explicitly: noncritical coordinates of the shearing vector correspond to x 1 , x 2 , x 17 , x 18 , x 19 .
13 The Lie superalgebra mb(4|5) over C In this section, we illustrate the algorithm presented in detail in [66], verify and rectify one formula from [17]. This algorithm allows one to describe vectorial Lie superalgebras by means of differential equations. In [64,65] the algorithm was used to describe the exceptional simple vectorial Lie superalgebras over C.
The Lie superalgebra mb(4|5) has three realizations as a transitive and primitive (i.e., not preserving invariant foliations on the space where it is realized by means of vector fields) vectorial Lie superalgebra. Speaking algebraically, the requirement that it should be transitive and primitive vectorial Lie superalgebra is the same as to have a W -filtration, so mb(4|5) has three W -filtrations.
Two of these W -filtrations are of depth 2, and one is of depth 3. In each realization this Lie superalgebra is the complete prolong of its negative part, see Section 2.11. In this section we consider the case of depth 3 (the grading K); i.e., we explicitly solve the differential equations singling out our Lie superalgebra. We thus explicitly obtain the expressions for the elements of mb(4|5; K).
In this realization, the Lie superalgebra g = mb(3|8) = mb(4|5; K) is the complete prolong of its negative part We would like to embed g − into the Lie superalgebra considered with the grading According to the algorithm described in [66], we find in v − two mutually commuting families of elements: X-vectors (the basis of g − ) and Y -vectors. The table of correspondences, where i = 1, 2, 3 and j = 1, 2: The nonzero commutation relations for the X-vectors are of the form ((i, j, k) ∈ A 3 ): The nonzero commutation relations for the Y -vectors correspond to the negative of the above structure constants: Let us represent an arbitrary vector field D ∈ vect(3|8) in the form As it was shown in [66], any X ∈ mb(3|8) is completely determined by a pair of functions F 1 , F 2 by means of equations, where i = 1, 2, 3 and (i, j, k) ∈ A 3 : (Comment: since (i, j, k) ∈ A 3 , i.e., is an even permutation, the formulas (13.4) and (13.5) are different; of course one can express the system by one formula inserting the sign of permutation.) Therefore the functions F 1 , F 2 must satisfy the following three groups of equations: The relations (13.2), (13.4) determine the remaining coordinates while the relations (13.3), (13.5) follow from (13.2), (13.4) and the commutation relations that the Y -vectors obey. Indeed, since Besides, We similarly get the expressions for the remaining coordinates: Therefore, an arbitrary element X ∈ mb(3|8) is of the form where the pair of functions F = {F 1 , F 2 } satisfies the system of equations (13.6). We select the Y -vectors so that the equations (13.6) the functions F 1 , F 2 should satisfy were as simple as possible. For example, take the following Y -vectors, where i = 1, 2, 3, s = 1, 2, (i, j, k) ∈ A 3 : Then, the corresponding X-vectors are of the form The Lie superalgebra mb(3|8) consists of the vector fields preserving the distribution determined by the following equations for the vector field D of the form (13.1): Let us express the coordinates f of the field D in the Y -basis in terms of the standard coordinates in the basis of partial derivatives: We get Therefore, in the standard coordinates, the distribution singled out by conditions (13.8) is given by the equations: The three equations determined by the first line of (13.9) allow one to express g u i and substitute into the third line to get Assuming that the pairing of the space of vector fields with that of 1-forms is given by the formula f ∂ ξ , gdξ = (−1) p(g) f g for any f, g ∈ F, we see that the distribution is singled out by Pfaff equations given by the following 1-forms 8 : Let us now solve the system (13.6).
Let us consider the last group of equations (13.6): To solve this system, take the expression (13.10) for F 2 and apply the operator Y η i . As a result, we get a function depending on various indeterminates, in particular, on ζ j . By virtue of (13.11), the coefficients of all monomials in ζ should vanish. Observe that the coefficient of ζ 1 ζ 2 ζ 3 vanishes automatically. The terms of degree 0 in ζ are of the form: Now, let us look at the degree 1 terms in ζ. To get them we should either take the term independent of ζ in expression (13.10) for F 2 (and this is α 2 ), and apply to it the degree 1 terms in ζ of Y η i , i.e., or, the other way round, take the degree 1 terms in ζ in (13.10), i.e., s ζ s ∂F 1 ∂ηs , and apply to it the degree 0 in ζ term of the operator Y η i , i.e., ∂ η i . Therefore, the terms of degree 1 in ζ are of the form: implying that So, the functions F 1 , F 2 are completely determined by the 5 functions α 1 , α 2 , f 1 , f 2 , f 3 that depend only on u and χ.
The terms of degree 2 in ζ follow from the same expression (13.10) and the same explanation as in the above paragraph leads to the equation (the coefficient of ζ j ζ k ): Let us expand this equation in parts corresponding to degrees in η. In degree 0 we have: In degrees 1, 2, 3 in η the equation (13.12) is automatically satisfied. Let us express equation (13.13) in the following more lucid way. We designate The equation (13.13) is equivalent to the following system of four equations: Let us describe the commutation relations in mb(3|8) more explicitly. Let us represent the vector field (13.7) as (13.14) and observe that, taking relation (13.2) and (13.4) into account, we have Observe that it suffices to compute only the defining components of F , G, and H: Then, we get In what follows we identify the vector field X F with the collection {α s , f i | s = 1, 2, i = 1, 2, 3}. (13.16) The bracket of vector fields corresponds to the bracket of such collections given by equations (13.15). Consider now the even part mb(3|8)0 of our algebra. Since p(F 1 ) = p(F 2 ) =1, it follows that p(α s ) =1 and p(f i ) =0 for all s and i. The component mb(3|8)0 has the three subspaces: The subspace V 1 is determined by the collection (13.16) such that Equations (13.15) imply that the vector fields generated by such functions form a commutative ideal in mb(3|8)0; we will identify this ideal with dΩ 1 (3): The subspace V 2 is determined by the collection (13.16) such that f i = 0 for i = 1, 2, 3. We will identify this space with Ω 0 (3) ⊗ sl(2), by setting where α ∈ Ω 0 (3), a, b, c ∈ C. Equations (13.15) imply that the subspaces V 1 and V 2 commute with each other whereas the brackets of two collections from V 2 is in our notation of the form Concerning V 3 , we have the following three natural ways to describe it: in all three cases we take f i = f i (u) for all i, whereas for the α s , we select one of the following: For p = 2, the case (c) is more convenient to simplify the brackets. Thus, we identify V 3 with vect(3), by means of the mapping The actions of D f on the subspace V 1 (as on the space of 2-forms) and V 2 (as on the space F ⊗ sl(2) of sl(2)-valued functions) are natural. The bracket of two elements if the form D f is, however, quite different from the usual bracket thanks to an extra term: Consider now the odd part: mb(3|8)1. We have p(F 1 ) = p(F 2 ) =0, and hence Let V 4 consist of collections (13.16) with f i = 0. We identify V 4 with Ω 0 (3) vol −1/2 ⊗C 2 , by setting Let V 5 consist of the collections (13.16), where We identify V 5 with Ω 2 (3) vol −1/2 ⊗C 2 , by assigning to the collection (13.18) the element Let us sum up a description of the spaces V i and their elements, see Table (13.19).
Having explicitly computed the brackets using expressions (13.15) and presenting the result by means of correspondences (13.19), we obtain the formulas almost identical to those offered in [17]. The difference, however, is vital: the Jacobi identity either holds or not.
We have already given the brackets of the even elements. The brackets of elements of mb0 and mb1 are of the form: To describe in these terms the bracket of two odd elements, perform the following natural identifications: {i,j,k}={1,2,3} such that (ijk)∈A 3 The bracket of two odd elements is of the form: In the last line above, the first summand lies in V 3 , the second one in V 2 , and the third one in V 1 . The difference as compared with [17]: the coefficient of the third summand in the last line on (13.20) should be 1 whereas in [17] it is equal to 1 2 . To verify, compute the Jacobi identity (it holds for 1 and not for 1 2 ) for the triple , where e 1 , e 2 span C 2 .
For p = 2, when case (c) in (13.17) is not defined, we can select any one of the cases (a) or (b), we take case (a) for definiteness. In these cases (a) and (b), we get two embeddings vect(3) ⊂ mb(3|8)0.
14 The Lie algebra F(mb (3; N |8)) is a true deform of svect 5; N In this section, we describe the analog of the complex Lie superalgebra mb(3|8) for p = 2 and consider its desuperization. For brevity, whenever possible we do not indicate the shearing vectors.
For consistency we replace χ i with u 3+i , and α i with f 3+i . Accordingly we denote Consider the mapping Clearly, this is a linear injective mapping. Formula (14.1) implies that ϕ(g) = svect (5). The mapping ϕ is not, however, an isomorphism of Lie algebras g and svect (5). Indeed, equations (13.15) rewritten in new notation imply the following equality (since p = 2, we skip the signs): Realization of kle convenient in what follows: for g = kle(5|10), we have g0 = svect(5|0) dΩ 3 and g1 = Π dΩ 1 with the natural g0-action on g1, while the bracket of any two odd elements is their product naturally identified with a divergence-free vector field.
For any D = 1≤i≤5 f i ∂ u i ∈ svect(5), we define and construct the embedding (as a vector space) Let us compute the bracket of two fields of the form (14.4): In order not to write too lengthy expressions, let us compute, separately, the brackets of individual summands. First, let i = 1, 2, 3, and k = 4, 5: Here we applied the Leibniz formula for the action of a vector field on a 2-form, and the expressions for the Lie derivative along the vector field X: Now, let k = 4 or 5: Finally, let i = 4 and k = 5: The expressions (14.5), (14.6), and (14.7) show that the through mapping ψ • ϕ determines an embedding g −→ kle, and hence the Lie algebra g is isomorphic to the thus-constructed Lie subalgebra of kle. is not a proper subalgebra of kle: it generates the whole kle. Indeed: take the bracket of the images of two fields of the form f ∂ 4 , g∂ 5 ∈ svect(5); we see, thanks to equation (14.7), that the image of svect(5) under the mapping (14.8) must contain 2-forms such as du 3 ∧ dh for certain h, and hence this image is not a subalgebra. Since the svect(5)-module dΩ 1 (5) is irreducible, the image of (14.8) generates the whole kle.
14.1 The Lie algebra F(mb (3; N |8)) is a true deform of svect 5; N Indeed, for the shearing vectors of the form N ∞ , all W-gradings of mb are the same as over C.
None of them has a maximal subalgebra of codimension 5, whereas svect(5) has such a subalgebra; cf. deforms described in [74,76] as well.
We consider g as a deform of svect (5) where is the usual bracket of vector fields, and the cocycle that determines the deform is All calculations in this realization are rather simple. We have (observe that thanks to formulas (14.2) and (14.9) brackets between the elements of g −1 are nontrivial, and g −1 generates the negative part) We also have 14.1.1 The deforms of svect(n; N ) for p > 3 These deforms are described in [76].

Partial prolongs
The Lie algebra g = F(mb(3|8)) constructed above is the complete prolong of its negative part, see Section 2.11; let us investigate if there is a partial prolong inside g. The component g 1 = V 1 ⊕ V 2 is the direct sum of the following g 0 -invariant subspaces: The g 0 -module V 1 is irreducible. The g 0 -module V 2 contains an irreducible g 0 -submodule V 0 2 = Span(x a x b ∂ i | a = b) ⊂ V 2 and g 0 acts in the quotient space as follows: sl(3) acts in V 2 /V 0 2 by zero and sl(2) acts as id sl(2) with multiplicity 3, so dim V 2 /V 0 2 = 8.
Using (14.9) it is easy to see that This means that only partial prolongs with g 1 ⊂ g 1 containing V 1 can be simple. For g 1 = V 1 ⊕ V 0 2 , the partial prolong with the unconstrained shearing vector which is of the form N u = (1, 1, 1, ∞, ∞) is a deform of svect 5; N u .
The subspace V 1 is commutative and the partial prolong h with V 1 as the first component is The simple 24-dimensional quotient obtained is isomorphic to sl(5) with the degrees of Chevalley generators being (0, ±1, 0, 0).
Conclusion. There are no new algebras as partial prolongs. Here vect(ξ) = Span f i (ξ)q i . Denote n := Λ(ξ)τ . Considering m −1 as a vect(ξ)-module, we preserve the multiplication of the Grassmann algebra Λ(ξ); i.e., the vect(ξ)-action satisfies the Leibniz rule, whereas the ideal n of m 0 does not preserve this multiplication. However, there is an isomorphism of vect(ξ)-modules σ : n −→ Π(m −1 ) and the action of n on m −1 is accomplished with the help of this isomorphism 9 : [f, g] = σ(f ) · g for any f ∈ n, g ∈ m −1 .
9 Speaking informally, although n does not preserve the multiplication in m−1 considered as the Grassmann algebra, n "remembers" this multiplication. And since m is the Cartan prolong, it also somehow "remembers" this structure.
The bilinear form ω with which we construct the central extension m− = m−2 ⊕ m−1 is the Berezin integral (the coefficient of the highest term) of the product of the two functions: ω(g1, g2) = g1g2 vol for any g1, g2 ∈ m−1; i.e., it also "remembers" the multiplication in m−1.
The "right" question therefore is not "which elements of m 0 preserve ω?", but rather "which elements of m 0 preserve ω conformally, up to multiplication by a scalar?" (15.1) It is precisely these elements which are derivations of the Lie superalgebra m − , and since m is the maximal algebra that "remembers" the multiplication, it follows that the whole of der(m − ) lies inside m 0 .
Let us give an interpretation of the analog of the space of semi-densities for p = 2.
For p = 2, we know the answer to the question (15.1): these are elements of the two types: (a) the elements of (b 1/2 (3)) 0 , the space of linear vector fields preserving the form ω, i.e., elements of the form (b) the elements of the form c · τ ∈ n which multiply ω by 2c ∈ K.
Thus, the form ω is preserved by svect(ξ) Λ(ξ) which is isomorphic to the subalgebra of linear (degree 0) vector fields in b ∞ . This should have been expected: since 2 = 0, then 1 2 = ∞. The elements conformally preserving ω are precisely ξ i ∂ i ←→ q i ξ i , so we have to add their sum to the 0th part and calculate the Cartan prolong. Now we are able to obtain the basis of the nonpositive components of mb(9; M ). A realization of the weight basis of the negative components and generators of the 0th component by vector fields is as follows, see Section 2.12 (X ± i are the Chevalley generators of sl(3) = svect(0|3) 0 ): For M unconstrained, dim g 1 = 64. The lowest-weight vectors in g 1 are  There are remarkable elements in g 1 : Each of the first three vectors generates a submodule of dim = 32; any two of the first three generate a submodule of dim = 40; all three together generate a submodule of dim = 48. The last one generates a submodule of dim = 8. All 4 together generate a submodule of g 1 of dim = 56. The quotient g 1 / g 1 is an irreducible g 0 -module. We have dim([g −1 , g 1 ]) = 25 while dim g 0 = 26; absent is the vector of weight 0: Note that [g −1 , g 1 ] = g 0 .
The elements absent in [g −1 , g 1i ] as compared with g 0 : Let us describe the complete prolong of this negative part of this Lie superalgebra, see Section 2.11. We deduce the form of the vector fields forming a basis of the negative part of mb(5; N |6) from nonzero commutation relations between ∂ k and x i ∂ a , where k = 1, . . . , 5, a = 3, 4, 5, and i = 1, 2, cf. (14.3), (14.9), considered as elements of F(mb(5; N |6)): For a basis we take realization in vector fields in 5 indeterminates z k , where k = 1, . . . , 5, and 6 indeterminates z ia , where a = 3, 4, 5 and i = 1, 2, of which z 1 , z 2 , z 3 , z 13 , z 23 are even while z 4 , z 5 , z 14 , z 15 , z 24 , z 25 are odd and δ i := ∂ z i : gi the generators (even | odd) g−1 δ1, δ2, δ13 + z1δ3, δ23 + z2δ3 | δ14 + z1δ4 + z25δ3, δ24 + z2δ4, δ15 + z1δ5 + z24δ3, δ25 + z2δ5 Because the bracket (16.1) is a deformation that does not preserve the grading given by the torus in gl (5), we consider the part of the weights that is salvaged, namely, we just exclude the 3rd coordinate of the weight; whereas the weight of x 3 is defined to be equal to (−1, −1, 1, 1). The dimension of g 0 is the same for all p; it is the expressions of the elements that differ. The raising operators in g 0 are those of weight (1, −1, 0, 0) or (0, 0, 1, −1), and those with a positive sum of coordinates of the weight, dim(g + 0 ) = 13; we skip their explicit description (it is commented with % marks in the T E X file available in arXiv).
17 On analogs of kas for p = 2 In this section, whenever possible, we do not indicate the shearing vectors. All computations in this section are performed for p = 2; however, for comparison, we also recall expressions obtained earlier over C in [64,65,67]. These expressions do not differ, usually, from those for p > 2. The Lie superalgebra kas over C was the last example needed to complete the list of simple W-graded vectorial Lie superalgebras, see [64,65]. Its nonpositive part is the same as that of g := k(1|6) (generated by the functions in the even t and 6 odd indeterminates) in its standard Z-grading while the component g 1 is exceptional, as a g 0 -module, among various k(1|n): only for n = 6 does g 1 split into 3 irreducible components: one depends on t, the other two are dual to each other. For any p = 2, we define two copies of kas; each of them is the partial prolong generated by the nonpositive part, and the two submodules of g 1 : the one that depends on t, and any one of the other two submodules.
To distinguish between these two isomorphic copies of kas, we denote by kas ξ the one whose space of generating functions contains the product ξ 1 ξ 2 ξ 3 ; let kas η be the one whose space of generating functions contains the product η 1 η 2 η 3 . We always consider only kas ξ , see (17.4), so we skip the superscript.
For p = 2, the structure of g 1 as a g 0 -module is rather complicated, and it is not clear what should one take for an analog of kas. Let us investigate.

Desuperization
Under desuperization k(1; N |6) turns into G := k 7; N , whereas the Lie algebra h, see (17.2), The highest-weight vectors of the G 0 -module S 3 (ξ, η) are as follows (in parentheses are the dimensions of the respective G 0 -modules these vectors generate) The lowest-weight vectors and the dimensions of the G 0 -modules these vectors generate are the same with the replacement ξ ←→ η. However, since the modules generated by lowest or highest-weight vectors do not span the whole of G 1 if p = 2, it is more natural to describe this component differently, as follows.
Bracketing ξ j ), and since each of the 26-dimensional modules generated by any cube contains only one cube, to have all squares in F(kas) 0 , we have to take for F(kas) 1 the module generated by all cubes. But we cannot do this: the prolong of the module containing all cubes is equal to k(7). Let us establish which cubes should be absent in the correct version of F(kas) 1 and how many versions are there.
At this stage we do not yet know what shall we eliminate in G 0 to get a correct version of F(kas) 0 , so we consider modules over g 0 .
is the smallest; observe that in E 3 (ξ, η) all squares vanish. By adding any of the following 8 one-dimensional modules spanned by expressions we can enlarge V and still have a g 0 -submodule. Together these modules span a 14-dimensional submodule W . The quotients E 3 (ξ, η)/W V * and W/V are irreducible g 0 -modules. Now, let us involve t. Set P := Span(tξ j , η i (t + ξ j η j ) | i = j). As is easy to see, dim V ∩ P = 3. The g 0 -module generated by tξ i is of dimension 16, as space it is the direct sum The g 0 -module generated by tξ i and tη j is of dimension 26; as a vector space it is the direct sum (V + P ) ⊕ the 8-dimensional space (17.3).
The module generated by ξ 1 is of dimension 16, and contains 5 elements with ξ 1 , V and 4 elements of the form ξ 1 x 2 x 3 , see (17.3). The intersection of all 6 such modules generated by cubes is equal to W = V ⊕the 8-dimensional space (17.3).
The dimension of the union of the modules generated by ξ j for all i and j is equal to 50, it is all G 1 except for V * . Verdict: where Y is the 8-dimensional module (17.3),
We have Clearly, see (17.1), We have and respectively we have
Finally, the tautological representation of o Π (6) is realized in the 6-dimensional quotient of Λ 3 (ξ, η); it cannot be, however, singled out as a SUBmodule, moreover, it is glued to the whole submodule U , including the elements X 0 and Y 0 . Now, look at the elements of g 1 whose expressions contain t. The Lie algebra h 0 = gl(3) acts in the same way as for p = 0 (as on the direct sum of the tautological gl(3)-module and its dual). Whereas Further, and {ξ i ξ j , ξ 2 (t + ξ 1 η 1 )} = 0 for all i, j.
Thus, under the action of h the space Q 1 = Span(tη 1 , tη 2 , tη 3 ) generates the space as well as V 2 , W 1 , and KX 0 . We can try to twist the elements tη i by adding something to them to enable the subalgebra h 2 annihilate them. Such twisted elements span the space P 1 := Span(η i (t + ξ j η j ) | i = j). Under the action of g −2 we obtain from P 1 the spaces P 2 := Span(tξ 1 , tξ 2 , tξ 3 ), W 2 , and KY 0 .  C A t with zero-diagonal symmetric matrices B and C and A ∈ sl (6) whereas h consists of the same type matrices with A ∈ gl(6). Therefore, g = kas(1; N |6) contains a simple ideal kas (1) (1; N |6) of codimension 1, its outer derivation being the outer derivation of g 0 . This derivation is present in the versions of kas considered in the next three sections.

Desuperizations of kas(1; N |6)
For one of the W-gradings of F(kas), we do not require presence of all squares in F(kas) 0 , but rather require their absence; this affects the number of parameters the shearing vector depends on.
Critical coordinates. The shearing vector N of the desuperization k := k 7; N , the ambient of the desuperized kas(1; N |6), has no critical coordinates.
For bases in g −1 and g 0 we take the following elements: Let k(1; N |6) be considered as preserving the distribution given by the form dt + ξ i dη i with the contact bracket (17.1) and the grading of the generating functions given by, see Table ( with vect(0|3) acting on g −1 as on the space of volume forms, i.e., D −→ D + div(D), and the element t generating the center of g 0 acts on g as the grading operator. To simplify notation, we redenote the indeterminates as follows:

Partial prolongs
The unconstrained shearing vector only depends on N 1 , we have sdim g 1 = 16|18.
The prolong in the direction of V 1 , see (6.2), is trivial, namely g (V 1 ) 2 = 0. The prolong in the direction of V 2 , see (6.2), gives sdim g (V 2 ) 2 = 4|4, and sdim g There are also 3 highest-weight vectors that generate nested modules W 1 ⊂ W 2 ⊂ W 3 ; we have The prolong in the direction of W 2 is trivial, as is the prolong in the direction of V 1 . The prolong in the direction of the 12|10-dimensional module V 2 + W 2 is equal to the prolong in the direction of V 2 .
The prolong in the direction of W 3 : sdim g (W 3 ) i = 12|12 for every i > 1; and N depends on one parameter: N 1 .
The prolong in the direction of the 16|15-dimensional module V 2 + W 3 is the same as for the whole of g 1 , and sdim g (V 2 +W 3 ) i = 16|16 for every i > 1; and hence N depends on one parameter: N 1 .
The lowest-weight vectors in g 1 are The highest-weight vectors in g 1 are

Desuperization
For the unconstrained shearing vector, we have dim g 1 = 55 with three lowest-weight vectors. The first two are as above, and the third one is The unconstrained shearing vector is of the form M = (m, 1, 1, 1, n, s, t).

Partial prolongs
For the unconstrained shearing vector, we have The unconstrained shearing vector for the prolong in the direction of V 3 depends on 1 parameter N 5 , and dim g (V 3 ) i = 32 for every i > 1. The unconstrained shearing vector for the prolong in the direction of V 3 + W 3 depends on 2 parameters N 1 , N 5 , and dim g 19 kas(8; M ) := F(kas(4; N |4)) Let k(1; N |6) be considered as preserving the distribution given by the form dt + ξ i dη i with the contact bracket (17.1) and the grading of the generating functions given by, see Table ( Indeed, the element (t + Φ)f (ξ) ∈ g 0 acts on g −1 as the operator of multiplication by f (ξ). Additionally g 0 contains the following operators: For p = 2, the last 3 elements in (19.1) do not belong to svect(0|3) and the elements (19.1) generate sl(1|3).
Critical coordinate: only N 1 .

Desuperization
The same as above, with dimension a + b instead of superdimension a|b.
20 The Lie superalgebra kas 5; N |5 ⊂ k(1; N |6; 1ξ) Whenever possible we do not indicate the shearing vector. Let k(1; N |6) be the Lie superalgebra which preserves the distribution given by the form dt+ ξ i dη i . Then, k(1; N |6) is endowed with the contact bracket (17.1); set deg K f = deg(f )−2, where the grading of the generating functions is given by . For a basis of the nonpositive part of g, we take the elements listed in (20.2).
To describe this component, we compare it with the complete prolong of the negative part, see Section 2.11. The 0th component of this prolong is equal to the 0th component of k(1; N |6; 1ξ). Its 3 elements that do not belong to g 0 are easy to find from the description of kas given in Section 17 (they are boxed): gi the basis elements The component g 0 contains two copies of sl(2); to distinguish them, we endow one of them with a tilde: sl(2) = sl(W ) generated by X + and X − , the other copy being sl(V ) generated by X + and X − . These two copies of sl(2) are "glued"; their glued sum has a common center spanned by E; i.e., their direct sum is factorized by a 1-dimensional subalgebra KZ in their 2-dimensional center, the explicit form of Z is inessential for us at the moment. Observe that D, ξ 1 ∂ ξ 1 / ∈ [g −1 , g 1 ]; only their sum D + ξ 1 ∂ ξ 1 ←→ ξ 1 η 1 + ξ 2 η 2 belongs to the commutant. In (20.2), we expressed the nonpositive part of g by means of vector fields in 10 indeterminates x setting ∂ i := ∂ x i .
The reader wishing to verify our computations will, of course, use the contact bracket and generating functions to compute inside g 0 . The realization by vector fields is only needed to compute g i for i > 0 (with computer's aid to speed up the process).
The only noncritical coordinate of the shearing vector N is N 2 ; it corresponds to what used to be t.
For the unconstrained shearing vector, we have sdim g 1 = 8|8. The only lowest-weight vector (w.r.t. the boxed operators) of g 1 that generates g 1 as a g 0 -module is The other lowest-weight vector and the only highest-weight vector (together and separately) generate a submodule V which, together with g −1 , generate an 8-dimensional part of g 0 . The quotient g 1 /V is an irreducible g 0 -module. Let us compute the bracket in (1 + Ξ)sb (1) (n; n) realized by elements of sb (1) (n). We have (1 + Ξ){f, g} if g = g i (q)ξ i and ∆(g) = 0; In the Z-grading of g = sb(n; n) by degrees of the q shifted by −1, we have: g −1 is spanned by monomials in ξ of degrees 1 through n − 1, and by 1 + Ξ; g 0 is spanned by functions of the form g = (1 + Ξ) g i (ξ)q i , where ∂ ξ i g i = 0.
The g 0 -action on g −1 is as follows. If deg ξ (g i ) > 0, then we can ignore Ξ in the factor 1 + Ξ since Ξ annihilates g i , and hence ad g acts on g −1 as the vector field g i ∂ ξ i acts on the space of functions in ξ.
If g = (1 + Ξ)q i , then the ad g -acts on g −1 precisely as an element of svect(0|n) acts on the space Vol ξ : Since (1 − Ξ) vol is the invariant subspace in Vol ξ , it follows that, in the quotient space, we can take for a basis elements of the form f (ξ) vol ξ , where monomials f differ from 1 and Ξ, and either 1 or Ξ. For reasons unknown, SuperLie selected Ξ, not 1. For p = 2, it is possible to desuperize deforms with odd parameters and consider them in the category of superspaces, see [12]. We assume that p(vol ξ ) ≡ n mod 2.

Example: n = 3
For a basis in g −1 , where ∂ i := ∂ x i , we take: For a basis of g 0 , where δ i := ∂ ξ i we take the following elements, where the g 0 -action on g −1 is given by realizations on the right of the ←→: operators, we could have considered a Z-grading of svect(0|n) by setting deg ξ n = −n + 1 and deg ξ 1 = · · · = deg ξ n−1 = 1 with ensuing natural division into "positive" and "negative" parts.) The highest-weight vectors of the g 0 -module g 1 are The lowest-weight vectors of the g 0 -module g 1 are Partial prolongs: The elements of g 0 absent in g 0 := [g 1 , g −1 ] are ξ 1 ξ 2 δ 3 , ξ 1 ξ 3 δ 2 , ξ 2 ξ 3 δ 1 . The g 0 -module g −1 is irreducible.
Let V i and W i denote the g 0 -modules generated by v i and w i , respectively. We have The brackets with g −1 : Therefore (recall the convention (6.2)) Partial prolongs in the direction of dimensions dim g1 = 10 absent are v3 and w3, dim g2 = 1, g3 = 0. 21.3 F( sb(2 n−1 − 1; N |2 n−1 )) for n even, p = 2 For the unconstrained shearing vector N u , the dimensions of homogeneous components of g = sb(2 n − 1; N u ) are the same as those of sb (1) (n) in the nonstandard grading sb (1) (n; n) for p = 0.

Partial prolongs
We have sdim g 0 = 25|24, and g 0 contains a simple ideal of sdim = 21|24, the quotient is commutative; g −1 is irreducible over this ideal. We have sdim g 1 = 56|55, there are 3 highestweight vectors and 2 lowest-weight vectors in g 1 ; The highest-weight vector of W 3 is w 3 = x (2) 4 ∂ 5 . This answer seems strange: the algebra is symmetric with respect to the permutation of the ξ i while the list of highest-weight vectors is not. Performing all possible permutations we obtain similar vectors x 3 ∂ 6 (which are not highest/lowest with respect to the division into positive/negative weights we have selected first), but generate similar submodules Other highest-weight vectors: (21. 2) The lowest-weight vectors: We have Partial prolongs of g 0 and the following parts of g 1 : from V 1 = W 1 and W 2 : sdim g 2 = 11|8, sdim g 3 = 0|1, no parameters; from V 2 : sdim g 2 = 33|32, 1 parameter: N 1 (same for W 3 , parameter N 4 ); from V 2 + W 3 : sdim g 2 = 56|56, 2 parameters: N 1 and N 4 , similar for Y i + Y j ; from Y 1 + Y 2 + Y 3 : sdim g 2 = 80|80, 3 parameters: N 1 , N 2 and N 3 .
Partial prolongs of the following parts of g 0 , see equation ( In this section, we can omit N when the arguments do not depend on it.

For p = 2
The first impression is that the characteristic-2 version of the Lie superalgebra vas does not exist: the cocycle that determines the central extension as of spe(4) is trivial, see [6]. The following problem is most natural.
Problem 22.1 (on analogs of as for p = 2). For p = 2, there are 8 analogs of pe(n) and 8 analogs of spe(n), and lots of their nontrivial central extensions, see [6]. Is there a nontrivial central extension e of one of these Lie (super)algebras, and an irreducible e-module M such that (M, e) 1 = 0?
We get an embedding g −→ vect(4|4). Let us describe the non-positive components of the embedded algebra. Let the coordinates of the ambient be x and ξ, and let us identify the basis elements of g −1 with the following vector fields in vect(4|4) = vect(x|ξ) Then, g −1 = Span{∂ y i , dy i } for i = 1, . . . , 4, and (g 0 )0 consists of the pairs (D, c), where D = i,j a ij y i ∂ y j is any vector field such that div D = 0, and c ∈ K, whereas (g 0 )1 consists of 1-forms y i dy j :
2) The critical coordinates of the shearing vector for the simple Lie algebra vas (1) 8; N -the desuperization of vas (1) (4; N |4) -are the ones that correspond to formerly odd indeterminates.

Partial prolongs
In order to investigate possible partial prolongs, we have to consider the g 0 -submodules V i of g 1 such that [vas −1 , V i ] = g 0 . Since it is not clear what is a lowest/highest-weight vector with respect to g 0 , we consider the lowest-weight vectors with respect to (g 0 )0, and build the g 0 -submodules from them.  1) Let V i be the g 0 -submodule of g 1 generated by v i , see (22.3). Then, [V i , g −1 ] = g 0 for all i.
3) In the quotient g 1 /V 1 , to each i ∈ {1, 2, 3, 4} there corresponds a 4|4-dimensional submodule M i spanned by the images of y (2) i ∂ y j and y (2) i dy j for j = 1, 2, 3, 4. Each M i is irreducible, and the images of M i and M j in g 1 /V 1 do not intersect for i = j. Thus, g 1 contains a submodule V 1 corresponding to N = 1, and up to four modules M i glued to V 1 if N = 1.
The partial prolongation in the direction of (⊕ i∈I⊂{1,2,3,4} M i ) V 1 is vas(4; N |4), where Idea of the proof . Since there is no complete reducibility, to prove item 3) we have to consider also highest-weight vectors (HWV) with respect to (g 0 )0. Then, we are able to find the two quotients modules M i invisible in  The algebra me(5; N ) is not simple, because Vol(2; N ) has a submodule of codimension 1; but me (1) (5; N ) is simple; in [22], Eick denoted what we denote me (1) (5; 1) by Bro 2 (1, 1). This algebra was discovered by Shen Guangyu, see [68], and should be denoted somehow to commemorate his wonderful discovery, we suggest to designate this Shen's analog of g(2) by gs (2).
There are two Z-gradings of g(2) with one pair of Chevalley generators of degree ±1 (the other generators being of degree 0): one Z-grading of depth 2 and the other one of depth 3. As is easy to see, for the grading of depth 3, the nonpositive parts of g(2) over fields K of characteristic p = 3 and those of me(5; N ) are isomorphic. Remarkably, this description holds for any p = 3, see [66]. For p = 3, the positive parts of the prolongation have the same dimensions as those of g(2) for p = 2, 5, but [g 1 , g −1 ] = K1 2 , the center of gl (2). (By the way, the realization of the nonpositive components of g(2), see equation (23.1), that works for p = 3, should be modified for p = 3, but we skip this since neither the complete prolong nor any partial prolong is simple.) Let U [k] be the gl(V )-module which is U as sl(V )-module, and let the central element z ∈ gl(V ) represented by the unit matrix, which acts on U [k] as k id, where k should be understood modulo p. Then, the grading of depth 3 is of the form Set ∂ i := ∂ x i to distinguish it from ∂ u i ; we use both representations in terms of x and u, whichever is more convenient. Here is the (borrowed from [66]) description of nonpositive components of me(5; N ), which are the same as those of gs(2) and g (2), by means of vector fields: gi the basis elements g−3 ∂u 1 ←→ ∂1, ∂u 2 ←→ ∂2 g−2 vol ←→ ∂3 g−1 X − 2 := u1 ←→ (x3 + x4x5)∂2 + ∂4, u2 ←→ x3∂1 + x4∂3 + ∂5 u1∂u 1 ←→ x1∂1 + x3∂3 + x4∂4, g0 X + 1 := u1∂u 2 ←→ x 5 ∂3 + x5∂4 gl(2) X − 1 := u2∂u 1 ←→ (x2 + x 4 ∂3 + x4∂5 u2∂u 2 ←→ x2∂2 + x3∂3 + x5∂5 Critical coordinates of me(5; N ): N 3 = 1. The g 0 -module g 1 is generated by the lowest-weight vector X + 2 ; we have dim g 1 = 2. Since X ± 1 and X + 2 contain x 4 and x 5 in degrees 2 and 3, see equation (23.1), the corresponding coordinates of the shearing vector in the generic case are ≥ 2; for the shearing vector with the smallest coordinates still ensuring simplicity; i.e., for N = (1, 1, 1, 2, 2), the prolong g is of dimension 17; it has ideals of dimension 14, 15, 16. The ideal of dimension 14 is simple, see [16,22,68]. do not lie in g (1) . In what follows we assume that k = n, for definiteness. As g is a sum of its Z n -weighted components, it suffices to show that D n cannot be obtained as the bracket of two elements homogenous with respect to the grading by the weight. As the x n -weight (i.e., weight with respect to x n ∂ n ) of D n is equal to −1, which is also the minimal possible x n -weight in g, it follows that, in order to obtain D n as a bracket, one of the factors (we say "factor" speaking about the Lie bracket, just as we do it for an associative multiplication) has to have weight −1 as well. Then, if this factor is homogenous w.r.t. the Z n -weight, it must be a monomial of the form a = 1≤i≤n−1 x (r i ) i ∂ n up to a scalar multiplier, where 0 ≤ r i < 2 N i . Then, from the weight considerations, the other factor must be of the form Clearly, So b ∈ g if and only if [a, b] = 0, hence g (1) contains no elements of the same weight as D n .

The Lie (super)algebra of contact vector fields
Let p = 2. As follows from equation (2.19), if 2n + 2 − m ≡ 0 mod p, then the Lie superalgebra k(2n + 1; N |m) is divergence-free, its derived algebra is simple.

On deforms of svect and h. Quantizations
In [74], Tyurin described non-isomorphic filtered deforms of the Lie algebras of series svect for p > 3 considered in the standard Z-grading. There are three statements in [74] that should be corrected.
First, in the introduction to [74], Tyurin wrote that in [32] Kac proved that all deforms of svect for p > 3 are filtered. Kac did not claim this in [32]. Moreover, Kac did not claim he described all filtered deformations, either; Kac writes only about filtered deformations associated with the standard Z-gradings.
Today, when the simple modular Lie algebras are classified for p > 3, the list of all their deforms is not needed for classification, but is a useful part of interpretation of the algebras found, see, e.g., [69,70]; this is of independent interest, like knowledge about "occasional isomorphisms" o(3) sl (2) or o(6) sl(4), or vect(1|1) m(1) k(1|2), as abstract Lie superalgebras.
Second, for any p, a particular deformation -called quantization in physical literatureof the Poisson Lie algebra on 2 indeterminates, induces a deform of svect(2; N ) h(2; N ), at least for N of the form (a, a) for any a ≥ 1, cf. [13]. Therefore, in [74], the claims describing all deformations of svect(m; N ) should have been confined to m > 2 and, moreover, Tyurin's main theorem should only claim a complete description of non-isomorphic filtered deforms related to the standard Z-grading; for examples of filtered deforms of svect (1) (3; 1) h (1) (4; 1) corresponding to distinct Z-gradings, see [18].
Although other deforms of h(2n; N ) do not provide us with new Lie algebras, they do provide us with new deforms, non-isomorphic to the filtered deforms.
Third, Wilson [76] corrected the main result of Tyurin who found all normal shapes of volume forms for p > 2, but missed an isomorphism. Wilson wrote only about normal shapes of volume forms, thus avoiding any discussion of deforms of svect.
The deform of svect(5; N ) we describe here is a completely new, exceptional, simple vectorial Lie algebra. It exists only for p = 2, the case neither Tyurin nor Wilson considered.
The characteristic-2 analogs of exceptional deformations of h and b described in [58] can have both even and odd parameters. The complete description of the deformations is unknown.

Remarks
In all lines Par N = dim N , except for the bottom line, see Section 21. To save space, we skip most of the conditions for simplicity in Table (25.2). In columns g i for i < 0, obviously, F is C or K. In lines N = 6, 7, we have λ = 2a n(a−b) for p = 2 and λ = a b for p = 2. In line 6 ∞ , we identify Vol 0 with a subspace of the space of functions F. In line 10, the Lie superalgebra svect µ (0|n) := (1 + µξ 1 · · · ξ n )svect(0|n) preserves the volume element (1 − µξ 1 · · · ξ n ) vol ξ , where p(µ) ≡ n mod 2.
For the notation C[i], see Section 2.1.1.

The exceptional simple vectorial Lie superalgebras over C as Cartan prolongs
For depth 2, we sometimes write (g −2 , g −1 , g 0 ) * for clarity. In Table ( To distinguish the two desuperizations of kle realized by vector fields on the spaces of the same dimension, we indicate by an index the depths of these algebras, e.g., kle 2 20; N ; if both algebras are of the same depth, we cover one of the desuperizations with a tilde. Clearly, under the desuperization we should ignore the change of parity in the negative components of F(g).