Classification of Finite Dimensional Modular Lie Superalgebras with Indecomposable Cartan Matrix

Finite dimensional modular Lie superalgebras over algebraically closed fields with indecomposable Cartan matrices are classified under some technical, most probably inessential, hypotheses. If the Cartan matrix is invertible, the corresponding Lie superalgebra is simple otherwise the quotient of the derived Lie superalgebra modulo center is simple (if its rank is greater than 1). Eleven new exceptional simple modular Lie superalgebras are discovered. Several features of classic notions, or notions themselves, are clarified or introduced, e.g., Cartan matrix, several versions of restrictedness in characteristic 2, Dynkin diagram, Chevalley generators, and even the notion of Lie superalgebra if the characteristic is equal to 2. Interesting phenomena in characteristic 2: (1) all simple Lie superalgebras with Cartan matrix are obtained from simple Lie algebras with Cartan matrix by declaring several (any) of its Chevalley generators odd; (2) there exist simple Lie superalgebras whose even parts are solvable. The Lie superalgebras of fixed points of automorphisms corresponding to the symmetries of Dynkin diagrams are also listed and their simple subquotients described.


Introduction
The ground field K is algebraically closed of characteristic p > 0. (Algebraic closedness of K is only needed in the quest for parametric families.) The term "Cartan matrix" will often be abbreviated to CM. Let the size of the square n × n matrix be equal to n.
Except the last section, we will only consider indecomposable Cartan matrices A with entries in K (so the integer entries are considered as elements of Z/pZ ⊂ K), indecomposability being the two conditions: 3. For the above listed Lie superalgebras g = g(A), their even parts g0 and the g0-modules g1 are explicitly described in terms of simple and solvable Lie algebras and modules over them.
4. For the above listed Lie superalgebras g = g(A), the existence of restrictedness is explicitly established; for p = 2, various cases of restrictedness are considered and explicit formulas given in each case.
Observe that although several of the exceptional examples were known for p > 2, together with one indecomposable Cartan matrix per each Lie superalgebra [17,18,13,14], the complete description of all inequivalent Cartan matrices for all the exceptional Lie superalgebras of the form g(A) and for ALL cases for p = 2 is new.
A posteriori we see that for each finite dimensional Lie superalgebra g(A) with indecomposable Cartan matrix, the module g1 is a completely reducible g0-module 1 .

Fixed points of the symmetries of Dynkin diagrams
We also listed the Lie superalgebras of fixed points of automorphisms corresponding to the symmetries of Dynkin diagrams and described their simple subquotients. In characteristic 0, this is the way all Lie algebras whose Dynkin diagrams has multiple bonds (roots of different lengths) are obtained. Since, for p = 2, there are no multiple bonds or roots of different length (at least, this notion is not invariant), it is clear that this is the way to obtain something new, although, perhaps, not simple. Lemma 2.2 in [23] implicitly describes the ideal in the Lie algebra of fixed points of an automorphism of a Lie algebra, but one still has to describe the Lie algebra of fixed points explicitly. This explicit answer is given in the last section. No new simple Lie (super)algebras are obtained.

On simple subquotients of Lie (super)algebras of the form g(A)
and a terminological problem Observe that if a given indecomposable Cartan matrix A is invertible, the Lie (super)algebra g(A) is simple, otherwise g(A) (i) /c -the quotient of its first (if i = 1) or second (if i = 2) derived algebra modulo the center -is simple (if sizeA > 1). Hereafter in this situation i = 1 or 2; meaning that the chain of derived algebras stabilizes (g(A) (j) g(A) (i) for any j ≥ i). We will see a posteriori that dim c = i = corank A, the maximal possible value of i above. This simple Lie algebra g(A) (i) /c does NOT possess any Cartan matrix. Except for Lie algebras over fields of characteristic p = 0, this subtlety is never mentioned causing confusion: The conventional sloppy practice is to refer to the simple Lie (super)algebra g(A) (i) /c as "possessing a Cartan matrix" (although it does not possess any) and at the same time to say "g(A) is simple" whereas it is not. However, it is indeed extremely inconvenient to be completely correct, especially in the cases where g(A), as well as its simple subquotient g(A) (i) /c, and a central extension of the latter, the algebra g(A) (i) , or all three at the same time might be needed. So we suggest to refer to either of these three algebras as "simple" ones, and extend the same convention to Lie superalgebras.
Thus, for i = 1, there are three distinct Lie (super)algebras: g(A) with Cartan matrix, g(A) (1) its derived, and g(A) (1) /c which is simple. It sometimes happens that the three versions are needed at the same time and to appropriately designate them is a non-trivial task dealt with in Section 4. When the second derived of g(A) is not isomorphic to the first one, i.e., when i = 2, there are even more intermediate derived objects, but fortunately only the three of them -g(A) with Cartan matrix, g(A) (2) , and g(A) (2) /c, the quotient modulo the whole center -appear (so far) in applications.

The Elduque Supermagic Square
For p > 2, Elduque interpreted most of the exceptional (when their exceptional nature was only conjectured; now this is proved) simple Lie superalgebras (of the form g(A)) in characteristic 3 [14] in terms of super analogs of division algebras and collected them into a Supermagic Square (an analog of Freudenthal's Magic Square). The rest of the exceptional examples for p = 3 and p = 5, not entering the Elduque Supermagic Square (the ones described here for the first time) are, nevertheless, somehow affiliated to the Elduque Supermagic Square [19].

Characteristic 2
Very interesting, we think, is the situation in characteristic 2. A posteriori we see that the list of Lie superalgebras in characteristic 2 of the form g(A) or g(A) (i) /c, where i = corank A can be equal to either 1 or 2, with an indecomposable matrix A is as follows: 1. Take any finite dimensional Lie algebra of the form g(A) with indecomposable Cartan matrix [54] and declare some of its Chevalley generators (simultaneously a positive one and the respective negative one) odd (the corresponding diagonal elements of A should be changed accordingly:0 to 0 and1 to 1, see Section 4.5). Let I be the vector of parities of the generators; the parity of each positive generator should equal to that of the corresponding negative one.
2. Do this for each of the inequivalent Cartan matrices of g(A) and any distribution I of parities.

If
A is not invertible, pass to g (i) (A, I)/c.

Such superization turns
1) a given orthogonal Lie algebra into either an ortho-orthogonal or a periplectic Lie superalgebra; 2) the three exceptional Lie algebras of e type turn into seven non-isomorphic Lie superalgebras of e type; 3) the wk type Lie algebras discovered in [54] turn into bgl type Lie superalgebras.
1.3. Remarks. 1) Observe that the Lie (super)algebra uniformly defined for any characteristic as • preserving a tensor, e.g., (super)trace-less; (ortho-)orthogonal and periplectic ones, • given by an integer Cartan matrix whose entries are considered modulo p may have different (super)dimensions as p varies (not only from 2 to "not 2"): The "same" Cartan matrices might define algebras of similar type but with different properties and names as the characteristic changes: for example sl(np) has CM in all characteristics, except p, in which case it is gl(np) that has a CM.
2) Although the number of inequivalent Cartan matrices grows with the size of A, it is easy to list all possibilities for serial Lie (super)algebras. Certain exceptional Lie superalgebras have dozens of inequivalent Cartan matrices; nevertheless, there are at least the following reasons to list all of them: 1. To classify all Z-gradings of a given g(A) (in particular, inequivalent Cartan matrices) is a very natural problem. Besides, sometimes the knowledge of the best, for the occasion, Z-grading is important. Examples of different cases: [44] (all simple roots non-isotropic), [38] (all simple roots odd); for computations "by hand" the cases where only one simple root is odd are useful.
In particular, the defining relations between the natural (Chevalley) generators of g(A) are of completely different form for inequivalent Z-gradings, and this is used in [44].
3. Certain properties of Cartan matrices may vary under the passage from one inequivalent CM to the other (the Lie superalgebras that correspond to such matrices may have different rate of growth as Z-graded algebras); this is a novel, previously unnoticed, feature of Lie superalgebras that had lead to false claims (rectified in [11]).

Related results
1) For explicit presentations in terms of (the analogs of) Chevalley generators of the Lie algebras and superalgebras listed here, see [2]. In addition to Serre-type relations there always are more complicated relations.
2) For deformations of the finite dimensional Lie (super)algebras of the form g(A) and g(A) (i) /c, see [6]. Observe that whereas "for p > 3, the Lie algebras with Cartan matrices of the same types that exist over C are either rigid or have deforms which also possess Cartan matrices", this is not so if p = 3 or 2.
3) For generalized Cartan-Tanaka-Shchepochkina (CTS) prolongs of the simple Lie (super)algebras of the form g(A), and the simple subquotients of such prolongs, see [5,7]. 4) With restricted Lie algebras one can associate algebraic groups; analogously, with restricted Lie superalgebras one can associate algebraic supergroups. For this and other results of Lebedev's Ph.D. thesis pertaining to the classification of simple modular Lie superalgebras, see [37].

.1 General notation
For further details on basics on Linear Algebra in Superspaces, see [39].
For the definition of the term "Lie superalgebra", especially, in characteristic 2, see Section 3. For the definition of the Lie (super)algebras of the form g(A), i.e., with Cartan matrix A (briefly referred to CM Lie (super)algebras), and simple relatives of the Lie superalgebras of the form g(A) with indecomposable Cartan matrix A, see Section 4. For simplicity of typing, the ith incarnation of the Lie (super)algebra g(A) with the ith CM (according to the lists given below) will be denoted by ig(A).
For the split form of any simple Lie algebra g, we denote the g-module with the ith fundamental weight π i by R(π i ) (as in [41,8]; these modules are denoted by Γ i in [24]). In particular, for o(2k +1), the spinor representation spin 2k+1 is defined to be the kth fundamental representation, whereas for o Π (2k), the spinor representations are the kth and the (k − 1)st fundamental representations. The realizations of these representations and the corresponding modules by means of quantization (as in [40]) can also be defined, since, as is easy to see, the same quantization procedure is well-defined for the restricted version of the Poisson superalgebra. (Fortunately, we do not need irreducible representations of o I (2k); their description is unknown, except the trivial, identity and adjoint ones.) In what follows, ad denotes both the adjoint representation and the module in which it acts. Similarly, id denotes both the identity (a.k.a. standard) representation of the linear Lie (super)algebra g ⊂ gl(V ) in the (super)space V and V itself. (We disfavor the adjective "natural" applied to id only, since it is no less appropriate to any of the tensor (symmetric, exterior) powers of id.) In particular, having fixed a basis in the n-dimensional space and having realized sl(V ) as sl(n), we write id instead of V , so V does not explicitly appear. The exterior powers and symmetric powers of the vector space V are defined as quotients of its tensor powers We set: where T . (V ) := ⊕T i (V ); let i (V ) and S i (V ) be homogeneous components of degree i.
The symbol A ⊂ + B denotes a semi-direct sum of modules of which A is a submodule; when dealing with algebras, A is an ideal in A ⊂ + B.

Superspaces
A superspace is a Z/2-graded space; for any superspace V = V0 ⊕ V1, where0 and1 are residues modulo 2; we denote by Π(V ) another copy of the same superspace: with the shifted parity, i.e., (Π(V ))ī = Vī +1 . The parity function is denoted by p. The superdimension of V is sdim V = a + bε, where ε 2 = 1, and a = dim V0, b = dim V1. (Usually, sdim V is shorthanded as a pair (a, b) or a|b; this notation obscures the fact that sdim V ⊗ W = sdim V · sdim W .) A superspace structure in V induces the superspace structure in the space End(V ). A basis of a superspace is always a basis consisting of homogeneous vectors; let Par = (p 1 , . . . , p dim V ) be an ordered collection of their parities. We call Par the format of (the basis of) V . A square supermatrix of format (size) Par is a sdim V × sdim V matrix whose ith row and ith column are of the same parity p i ∈ Par. The matrix unit E i,j is of parity p i + p j . Whenever possible, we consider one of the simplest formats Par, e.g., the format Par st of the form (0, . . . ,0;1, . . . ,1) is called standard. Systems of simple roots of Lie superalgebras corresponding to distinct nonstandard formats of supermatrix realizations of these superalgebras are related by so-called odd reflections.

The Sign Rule
The formulas of Linear Algebra are superized by means of the Sign Rule: if something of parity p moves past something of parity q, the sign (−1) pq accrues; the expressions defined on homogeneous elements are extended to arbitrary ones via linearity.
Examples of application of the Sign Rule: By setting we get the notion of the supercommutator and the ensuing notions of supercommutative and superanti-commutative superalgebras; Lie superalgebra is the one which, in addition to superanticommutativity, satisfies Jacobi identity amended with the Sign Rule; a superderivation of a given superalgebra A is a linear map D : A −→ A satisfying the super Leibniz rule Let V be a superspace, sdim V = m|n. The general linear Lie superalgebra of operators acting in V is denoted by gl(V ) or gl(m|n) if an homogeneous basis of V is fixed. The Lie subsuperalgebra of supertraceless operators (supermatrices) is denoted sl(V ) sl(m|n). If a Lie superalgebra g ⊂ gl(V ) contains the ideal of scalar matrices s = K1 m|n , then pg = g/s denotes the projective version of g.
Observe that sometimes the Sign Rule requires some dexterity in application. For example, we have to distinguish between super-skew and super-anti although both versions coincide in the non-super case: In other words, "anti" means the total change of the sign, whereas any "skew" notion can be straightened by the parity change. In what follows, the supersymmetric bilinear forms and supercommutative superalgebras are named according to the above definitions. The supertransposition of supermatrices is defined so as to assign the supertransposed supermatrix to the dual operator in the dual bases. For details, see [36]; an explicit expression in the standard format is as follows (we give a general formula for matrices with entries in a supercommutative superalgebra; in this paper we only need the lower formula): for X odd.

What the Lie superalgebra is
Dealing with superalgebras it sometimes becomes useful to know their definition. Lie superalgebras were distinguished in topology in 1930's, and the Grassmann superalgebras half a century earlier. So it might look strange when somebody offers a "better" definition of a notion which was established about 70 year ago. Nevertheless, the answer to the question "what is a (Lie) superalgebra?" is still not a common knowledge. So far we defined Lie superalgebras naively: via the Sign Rule. However, the naive definition suggested above ("apply the Sign Rule to the definition of the Lie algebra") is manifestly inadequate for considering the supervarieties 2 of deformations and for applications of representation theory to mathematical physics, for example, in the study of the coadjoint representation of the Lie supergroup which can act on a supermanifold or supervariety but never on a vector superspace -an object from another category. We were just lucky in the case of finite dimensional Lie algebras over C that the vector spaces can be viewed as manifolds or varieties. In the case of spaces over K and in the super setting, to be able to deform Lie (super)algebras or to apply group-theoretical methods, we must be able to recover a supermanifold from a vector superspace, and vice versa.
A proper definition of Lie superalgebras is as follows. The Lie superalgebra in the category of supervarieties corresponding to the "naive" Lie superalgebra L = L0 ⊕ L1 is a linear supermanifold L = (L0, O), where the sheaf of functions O consists of functions on L0 with values in the Grassmann superalgebra on L * 1 ; this supermanifold should be such that for "any" (say, finitely generated, or from some other appropriate category) supercommutative superalgebra C, the space L(C) = Hom(Spec C, L), called the space of C-points of L, is a Lie algebra and the correspondence C −→ L(C) is a functor in C. (A. Weil introduced this approach in algebraic geometry in 1953; in super setting it is called the language of points or families.) This definition might look terribly complicated, but fortunately one can show that the correspondence L ←→ L is one-to-one and the Lie algebra L(C), also denoted L(C), admits a very simple description: A Lie superalgebra homomorphism ρ : L 1 −→ L 2 in these terms is a functor morphism, i.e., a collection of Lie algebra homomorphisms ρ C : L 1 (C) −→ L 2 (C) such that any homomorphism of supercommutative superalgebras ϕ : C −→ C 1 induces a Lie algebra homomorphism ϕ : L(C) −→ L(C 1 ) and products of such homomorphisms are naturally compatible. In particular, a representation of a Lie superalgebra L in a superspace V is a homomorphism ρ : L −→ gl(V ), i.e., a collection of Lie algebra homomorphisms ρ C : L(C) −→ (gl(V ) ⊗ C)0.
2.4.1. Example. Consider a representation ρ : g −→ gl(V ). The space of infinitesimal deformations of ρ is isomorphic to H 1 (g; V ⊗ V * ). For example, if g is the 0|n-dimensional (i.e., purely odd) Lie superalgebra (with the only bracket possible: identically equal to zero), its only irreducible modules are the trivial one, 1, and Π(1). Clearly, and, because the Lie superalgebra g is commutative, the differential in the cochain complex is zero. Therefore H 1 (g; 1) = 1 (g * ) g * , so there are dim g odd parameters of deformations of the trivial representation. If we consider g "naively", all of these odd parameters will be lost.
Examples that lucidly illustrate why one should always remember that a Lie superalgebra is not a mere linear superspace but a linear supermanifold are, e.g., the deforms with odd parameters. In the category of supervarieties, these deforms, listed in [6], are simple Lie superalgebras.
3 What the Lie superalgebra in characteristic 2 is (from [34]) Let us give a naive definition of a Lie superalgebra for p = 2. We define it as a superspace g = g0 ⊕ g1 such that g0 is a Lie algebra, g1 is an g0-module (made into the two-sided one by anti-symmetry, but if p = 2, it is the same) and on g1 a squaring (roughly speaking, the halved bracket) is defined as a map x → x 2 such that (ax) 2 = a 2 x 2 for any x ∈ g1 and a ∈ K, and (x + y) 2 − x 2 − y 2 is a bilinear form on g1 with values in g0.
(3.1) (We use a minus sign, so the definition also works for p = 2.) The origin of this operation is as follows: If char K = 2, then for any Lie superalgebra g and any odd element x ∈ g1, we have If p = 2, we define x 2 first, and then define the bracket of odd elements to be (this equation is valid for p = 2 as well): We also assume, as usual, that if x, y ∈ g0, then [x, y] is the bracket on the Lie algebra; if x ∈ g0 and y ∈ g1, then [x, y] := l x (y) = −[y, x] = −r x (y), where l and r are the left and right g0-actions on g1, respectively.
The Jacobi identity involving odd elements has now the following form: for any x ∈ g1, y ∈ g. If K = Z/2Z, we can replace the condition on two odd elements by a simpler one: Because of the squaring, the definition of derived algebras should be modified. For any Lie superalgebra g, set g (0) := g and 3.1 Examples: Lie superalgebras preserving non-degenerate forms [34] Lebedev investigated various types of equivalence of bilinear forms for p = 2, see [34]; we just recall the verdict and say that two (anti)-symmetric bilinear forms B and B on a superspace V are equivalent if there is an even non-degenerate linear map M : V → V such that We fix some basis in V and identify a given bilinear form with its Gram matrix in this basis; let us also identify any linear operator on V with its matrix. Then two bilinear forms (rather supermatrices) are equivalent if there is an even invertible matrix M such that We often use the following matrices Let J n|n and Π n|n be the same as J 2n and Π 2n but considered as supermatrices.
Lebedev proved that, with respect to the above natural 3 equivalence of forms (3.7), every even symmetric non-degenerate form on a superspace of dimension n0|n1 over a perfect field of characteristic 2 is equivalent to a form of the shape (here: i =0 or1 and each n i may equal to 0) In other words, the bilinear forms with matrices 1 n and Π n are equivalent if n is odd and nonequivalent if n is even; antidiag(1, . . . , 1) ∼ Π n for any n. The precise statement is as follows: 3.1.1. Theorem. Let K be a perfect field of characteristic 2. Let V be an n-dimensional space over K.
1) For n odd, there is only one equivalence class of non-degenerate symmetric bilinear forms on V .
2) For n even, there are two equivalence classes of non-degenerate symmetric bilinear forms, one -with at least one non-zero element on the main diagonal -contains 1 n and the other onewith only 0s on the main diagonal -contains S n := antidiag(1, . . . , 1) and Π n .
The Lie superalgebra preserving B -by analogy with the orthosymplectic Lie superalgebras osp in characteristic 0 we call it ortho-orthogonal and denote oo B (n0|n1) -is spanned by the supermatrices which in the standard format are of the form Since, as is easy to see, we do not have to consider the Lie superalgebra oo ΠI (n0|n1) separately unless we study Cartan prolongations -the case where the difference between these two incarnations of the same algebra is vital. For an odd symmetric form B on a superspace of dimension (n0|n1) over a field of characteristic 2 to be non-degenerate, we need n0 = n1, and every such form B is equivalent to Π k|k , where k = n0 = n1. This form is preserved by linear transformations with supermatrices in the standard format of the shape where A ∈ gl(k), C and D are symmetric k × k matrices. (3.9) The Lie superalgebra pe(k) of supermatrices (3.9) will be referred to as periplectic, as A. Weil suggested, and denoted by pe B (k) or just pe(k). (Notice that the matrix realization of pe B (k) over C or R is different: its set of roots is not symmetric relative the change of roots "positive ←→ negative".) The fact that two bilinear forms are inequivalent does not, generally, imply that the Lie (super)algebras that preserve them are not isomorphic. (3.10) In [34], Lebedev proved that for the non-degenerate symmetric forms, this implication (3.10) is, however, true (except for oo IΠ (n0|n1) oo ΠI (n1|n0) and oo (1) ΠΠ (6|2) pe (1) (4)) and described the distinct types of Lie (super)algebras preserving non-degenerate forms. In what follows, we describe which of these Lie (super)algebras (or their derived ones) are simple, and which of them (or their central extensions) "have Cartan matrix". But first, we recall what does the term in quotation marks mean. For the most reasonable definition of Lie algebra with Cartan matrix over C, see [28]. The same definition applies, practically literally, to Lie superalgebras and to modular Lie algebras and to modular Lie superalgebras. However, the usual sloppy practice is to attribute Cartan matrices to many of those (usually simple) modular Lie algebras and (modular or not) Lie superalgebras which, strictly speaking, have no Cartan matrix! Although it may look strange for the reader with non-super experience over C, neither the simple modular Lie algebra psl(pk), nor the simple modular Lie superalgebra psl(a|pk + a), nor -in characteristic 0 -the simple Lie superalgebra psl(a|a) possesses a Cartan matrix. Their central extensions -sl(pk), the modular Lie superalgebra sl(a|pk + a), and -in characteristic 0the Lie superalgebra sl(a|a) -do not have Cartan matrix, either.
Their relatives possessing a Cartan matrix are, respectively, gl(pk), gl(a|pk + a), and gl(a|a), and for the "extra" (from the point of view of sl or psl) grading operator (such operators are denoted in what follows d i , to distinguish them from the "inner" grading operators h i ) we take E 1,1 .
Since often all the Lie (super)algebras involved (the simple one, its central extension, the derivation algebras thereof) are needed (and only representatives of one of the latter types of Lie (super)algebras are of the form g(A)), it is important to have (preferably short and easy to remember) notation for each of them.
For the Lie algebras that preserve a tensor (bilinear form or a volume element) we retain the same notation in all characteristic; but the (super)dimension of various incarnations of the algebras with the same name may differ as characteristic changes; simplicity may also be somewhat spoilt.
For the Lie (super)algebras which are easiest to be determined by their Cartan matrix (the Elduque Supermagic Square is of little help here), we have: for p = 3: e(6) is of dimension 79, its derived e(6) (1) is of dimension 78, whereas its "simple core" is e(6) (1) /c of dimension 77; g(2) is not simple (moreover, its CM is decomposable); its "simple core" is isomorphic to psl(3); for p = 2: e(7) is of dimension 134, its derived e(7) (1) is of dimension 133, whereas its "simple core" is e(7) (1) /c of dimension 132; g(2) constructed from its CM reduced modulo 2 is now isomorphic to gl(4); it is not simple, its "simple core" is isomorphic to psl(4); the orthogonal Lie algebras and their super analogs are considered in detail later.
In our examples, the notation D/d|B means that sdim g(A) = D|B whereas sdim g(

Generalities
Let us start with the construction of a CM Lie (super)algebra. Let A = (A ij ) be an n×n-matrix.
Let rk A = n − l. It means that there exists an l × n-matrix T = (T ij ) such that a) the rows of T are linearly independent; b) T A = 0 (or, more precisely, "zero l × n-matrix"). Indeed, if rk A T = rk A = n − l, then there exist l linearly independent vectors v i such that Let the elements e ± i and h i , where i = 1, . . . , n, generate a Lie superalgebra denotedg(A, I), where I = (p 1 , . . . p n ) ∈ (Z/2) n is a collection of parities (p(e ± i ) = p i ), free except for the relations Let Lie (super)algebras with Cartan matrix g(A, I) be the quotient ofg(A, I) modulo the ideal we explicitly described in [3,4,2]. By abuse of notation we retain the notations e ± j and h j -the elements ofg(A, I) -for their images in g(A, I) and g (i) (A, I).
The additional to (4.3) relations that turng(A, I) into g(A, I) are of the form R i = 0 whose left sides are implicitly described, for the general Cartan matrix with entries in K, as the R i that generate the ideal r maximal among the ideals ofg(A, I) whose intersection with the span of the above h i and the d j described in equation (4.8) is zero.
where i = 1, . . . , l. (4.6) The existence of central elements means that the linear span of all the roots is of dimension n − l only. (This can be explained even without central elements: The weights can be considered as column-vectors whose i-th coordinates are the corresponding eigenvalues of ad h i . The weight of e i is, therefore, the i-th column of A. Since rk A = n − l, the linear span of all columns of A is (n − l)-dimensional just by definition of the rank. Since any root is an (integer) linear combination of the weights of the e i , the linear span of all roots is (n − l)-dimensional.) This means that some elements which we would like to see having different (even opposite if p = 2) weights have, actually, identical weights. To fix this, we do the following: Let B be an arbitrary l × n-matrix such that the (n + l) × n-matrix A B has rank n. Let us add to the algebra g =g(A, I) (and hence g(A, I)) the grading elements d i , where i = 1, . . . , l, subject to the following relations: (the last two relations mean that the d i lie in the Cartan subalgebra, and even in the maximal torus which will be denoted by h).
Note that these d i are outer derivations of g(A, I) (1) , i.e., they can not be obtained as linear combinations of brackets of the elements of g(A, I) (i.e., the d i do not lie in g(A, I) (1) ).

Roots and weights
In this subsection, g denotes one of the algebras g(A, I) org(A, I).
Let h be the span of the h i and the d j . The elements of h * are called weights. For a given weight α, the weight subspace of a given g-module V is defined as Any non-zero element x ∈ V is said to be of weight α. For the roots, which are particular cases of weights if p = 0, the above definition is inconvenient because it does not lead to the modular analog of the following useful statement.

Statement ([28]
). Over C, the space of any Lie algebra g can be represented as a direct sum of subspaces Note that if p = 2, it might happen that h g 0 . (For example, all weights of the form 2α over C become 0 over K.) To salvage the formulation of Statement in the modular case with minimal changes, at least for the Lie (super)algebras g with Cartan matrix -and only this case we will have in mind speaking of roots, we decree that the elements e ± i with the same superscript (either + or −) correspond to linearly independent roots α i , and any root α such that g α = 0 lies in the Z-span of {α 1 , . . . , α n }, i.e., g = α∈Z{α 1 ,...,αn} g α . (4.10) Thus, g has a R n -grading such that e ± i has grade (0, . . . , 0, ±1, 0, . . . , 0), where ±1 stands in the i-th slot (this can also be considered as Z n -grading, but we use R n for simplicity of formulations). If p = 0, this grading is equivalent to the weight grading of g. If p > 0, these gradings may be inequivalent; in particular, if p = 2, then the elements e + i and e − i have the same weight. (That is why in what follows we consider roots as elements of R n , not as weights.) Any non-zero element α ∈ R n is called a root if the corresponding eigenspace of grade α (which we denote g α by abuse of notation) is non-zero. The set R of all roots is called the root system of g.
Clearly, the subspaces g α are purely even or purely odd, and the corresponding roots are said to be even or odd.

Systems of simple and positive roots
In this subsection, g = g(A, I), and R is the root system of g.
For any subset B = {σ 1 , . . . , σ m } ⊂ R, we set (we denote by Z + the set of non-negative integers): The set B is called a system of simple roots of R (or g) if σ 1 , . . . , σ m are linearly independent and R = R + B ∪ R − B . Note that R contains basis coordinate vectors, and therefore spans R n ; thus, any system of simple roots contains exactly n elements.
A subset R + ⊂ R is called a system of positive roots of R (or g) if there exists x ∈ R n such that (Here (·, ·) is the standard Euclidean inner product in R n ). Since R is a finite (or, at least, countable if dim g(A) = ∞) set, so the set is a finite/countable union of (n − 1)-dimensional subspaces in R n , so it has zero measure. So for almost every x, condition (4.11) holds. By construction, any system B of simple roots is contained in exactly one system of positive roots, which is precisely R + B . 4.4.1. Statement. Any finite system R + of positive roots of g contains exactly one system of simple roots. This system consists of all the positive roots (i.e., elements of R + ) that can not be represented as a sum of two positive roots.
We can not give an a priori proof of the fact that each set of all positive roots each of which is not a sum of two other positive roots consists of linearly independent elements. This is, however, true for finite dimensional Lie algebras and Lie superalgebras of the form g(A) if p = 2.

Normalization convention
Clearly, Two pairs (A, I) and (A , I ) are said to be equivalent if (A , I ) is obtained from (A, I) by a composition of a permutation of parities and a rescaling A = diag(λ 1 , . . . , λ n ) · A, where λ 1 · · · λ n = 0. Clearly, equivalent pairs determine isomorphic Lie superalgebras.
The rescaling affects only the matrix A B , not the set of parities I B . The Cartan matrix A is said to be normalized if A jj = 0 or 1, or 2. (4.13) We let A jj = 2 only if i j =0; in order to distinguish between the cases where i j =0 and i j =1, we write A jj =0 or1, instead of 0 or 1, if i j =0. We will only consider normalized Cartan matrices; for them, we do not have to describe I. The row with a 0 or0 on the main diagonal can be multiplied by any nonzero factor; usually (not only in this paper) we multiply the rows so as to make A B symmetric, if possible.

Equivalent systems of simple roots
Let B = {α 1 , . . . , α n } be a system of simple roots. Choose non-zero elements e ± i in the 1dimensional (by definition) superspaces It would be nice to find a convenient way to fix some distinguished pair (A B , I B ) in the equivalence class. For the role of the "best" (first among equals) order of indices we propose the one that minimizes the value |i − j| (4.14) (i.e., gather the non-zero entries of A as close to the main diagonal as possible). Observe that this numbering differs from the one that Bourbaki use for the e type Lie algebras.

Chevalley generators and Chevalley bases
We often denote the set of generators corresponding to a normalized matrix by X ± 1 , . . . , X ± n instead of e ± 1 , . . . , e ± n ; and call them, together with the elements H i := [X + i , X − i ], and the derivatives d j added for convenience for all i and j, the Chevalley generators.
For p = 0 and normalized Cartan matrices of simple finite dimensional Lie algebras, there exists only one (up to signs) basis containing X ± i and H i in which A ii = 2 for all i and all structure constants are integer, cf. [51]. Such a basis is called the Chevalley basis.
Observe that, having normalized the Cartan matrix of o(2n + 1) so that A ii = 2 for all i = n but A nn = 1, we get another basis with integer structure constants. We think that this basis also qualifies to be called Chevalley basis; for Lie superalgebras, and if p = 2, such normalization is a must.
Conjecture. If p > 2, then for finite dimensional Lie (super)algebras with indecomposable Cartan matrices normalized as in (4.13), there also exists only one (up to signs) analog of the Chevalley basis.
We had no idea how to describe analogs of Chevalley bases for p = 2 until recently; clearly, the methods of the recent paper [12] should solve the problem.

Restricted Lie superalgebras
Let g be a Lie algebra of characteristic p > 0. Then, for every x ∈ g, the operator ad p x is a derivation of g. If it is an inner derivation for every x ∈ g, i.e., if ad p x = ad x [p] for some element denoted x [p] , then the corresponding map is called a p-structure on g, and the Lie algebra g endowed with a p-structure is called a restricted Lie algebra. If g has no center, then g can have not more than one p-structure. The Lie algebra gl(n) possesses a p-structure, unique up to the contribution of the center; this p-structure is used in the next definition. The notion of a p-representation is naturally defined as a linear map ρ : [p] ); in this case V is said to be a p-module.
Passing to superalgebras, we see that, for any odd D ∈ derA, we have So, if char K = p, then D 2p is always an even derivation for any odd D ∈ derA. Now, let g be a Lie superalgebra of characteristic p > 0. Then for every x ∈ g0, the operator ad p x is a derivation of g, i.e., g0-action on g1 is a p-representation; for every x ∈ g1, the operator ad 2p for any x ∈ g1. We demand that for any x ∈ g0, we have as operators on the whole g, i.e., g1 is a restricted g0-module.
Then the pair of maps is called a p|2p-structure -or just p-structure -on g, and the Lie superalgebra g endowed with a p-structure is called a restricted Lie superalgebra.

The case where g0 has center
The p-structure on g0 does not have to determine a p|2p-structure on g: Even if the actions of ad p x and ad x [p] coincide on g0, they do not have to coincide on the whole of g. This remark affects even simple Lie superalgebras if g0 has center. We can not say if a p-structure on g0 defines a p|2p-structure on g in the case of centerless g0: To define it we need to have, separately, a p-module structure on g1 over g0.
For the case where the Lie superalgebra g or even g0 has center, the following definition is more appropriate: g is said to be restricted if there is given the map such that, for any a ∈ K, we have for an indeterminate a,
5.1.1. Remark. If g is centerless, we do not need conditions 1) and 2) of (5.4) since they follow from 3).

Proposition.
1) If p > 2 (or p = 2 but A ii =1 for all i) and g(A) is finite-dimensional, then g(A) has a p|2p-structure such that 2) If all the entries of A are elements of Z/pZ, then we can set h Proof . For the simple Lie algebras, the p-structure is unique if any exists, see [27]. The same proof applies to simple Lie superalgebras and p|2p-structures. The explicit construction completes the proof of headings 1) and 2). To prove 3) a counterexample suffices; we leave it as an exercise to the reader to produce one.

Remarks.
1) It is not enough to define p|2p-structure on generators, one has to define it on a basis.
For examples of simple Lie superalgebras without Cartan matrix but with a p|2p-structure, see [37]. In addition to the expected examples of the modular versions of Lie superalgebras of vector fields, and the queer analog of the gl series, there are -for p = 2 -numerous (and hitherto unexpected) queerifications, see [37].

(2, 4)-structure on Lie algebras
If p = 2, we encounter a new phenomenon first mentioned in [33]. Namely, let g = g + ⊕ g − be a Z/2-grading of a Lie algebra. We say that g has a (2, −)-structure, if there is a 2-structure on g + but not on g. It sometimes happens that this (2, −)-structure can be extended to a 4 (2, 4)structure, which means that for any x ∈ g − there exists an x [4] ∈ g + such that ad 4 x = ad x [4] .
(2, 4|2)-structure on Lie superalgebras. A generalization of the (2, 4)-structure from Lie algebras to Lie superalgebras (such as oo (1) IΠ (2n + 1|2m)) is natural: Define the Z/2-grading g = g + ⊕ g − of a Lie superalgebra having nothing to do with the parity similarly to that of g(A) = o (1) (2n + 1), and define the squaring on the plus part and a (2, 4)-structure on the minus part such that the conditions ad x [2] (y) = ad 2 x (y) for all x ∈ (oo (1) are satisfied for any y ∈ oo  (2|2)-structure on Lie superalgebras. Lebedev observed that if p = 2 and a Lie superalgebra g possesses a 2|4-structure, then the Lie algebra F (g) one gets from g by forgetting the superstructure (this is possible since [x, x] = 2x 2 = 0 for any odd x) possesses a 2-structure given by the "2" part of 2|4-structure on the former g0; the squaring on g1; the rule (x + y) [2] = x [2] + y [2] + [x, y] on the formerly inhomogeneous (with respect to parity) elements.
So one can say that if p = 2, then any restricted Lie superalgebra g (i.e., the one with a 2|4-structure) induces a 2|2-structure on the Lie algebra F (g) which is defined even on inhomogeneous elements (unlike p|2p-structures defined on homogeneous elements only).

5.2.2.1.
Remark. The restricted Lie superalgebra structures resemble (somehow) a hidden supersymmetry of the following well-known fact: The product of two vector fields is not necessarily a vector field, whereas their commutator always is a vector field. (5.8) This fact was not considered to be supersymmetric until recently: Dzhumadildaev investigated a similar phenomenon: For the general and divergence-free Lie algebras of polynomial vector fields in n indeterminates over C, he investigated for which N = N (n) the anti-symmetrization of the map D −→ D N (i.e., the expression ) yields a vector field. For the answer for n = 2, 3 and a conjecture, see [16]. But the most remarkable is Dzhumadildaev's discovery of a hidden supersymmetry of the usual commutator described by a universal odd vector field. Dzhumadildaev deduced the above fact (5.8) from the following property of odd vector fields: The product of two vector fields is not necessarily a vector field, whereas the square of any odd field always is a vector field. (5.9)

Non-degenerate even supersymmetric bilinear forms and ortho-orthogonal Lie superalgebras
For p = 2, there are, in general, four equivalence classes of inequivalent non-degenerate even supersymmetric bilinear forms on a given superspace. Any such form B on a superspace V of superdimension n0|n1 can be decomposed as follows: where B0, B1 are symmetric non-degenerate forms on V0 and V1, respectively. For i =0,1, the form B i is equivalent to 1 n i if n i is odd, and equivalent to 1 n i or Π n i if n i is even. So every non-degenerate even symmetric bilinear form is equivalent to one of the following forms (some of them are defined not for all dimensions): We denote the Lie superalgebras that preserve the respective forms by oo II (n0|n1), oo IΠ (n0|n1), oo ΠI (n0|n1), oo ΠΠ (n0|n1), respectively. Now let us describe these algebras.
If n := n0 + n1 ≥ 3, then the Lie superalgebra oo (1) II (n0|n1) is simple. This Lie superalgebra has a 2|4-structure; it has no Cartan matrix. 6.1.1.1. Remark. To prove that a given Lie (super)algebra g has no Cartan matrix, we have to consider its maximal tori, and for each of them, take the corresponding root grading. Then, if the simple roots are impossible to define, or the elements of weight 0 do not commute, etc. -if any of the requirements needed to define the Lie (super)algebra with Cartan matrix is violatedwe are done. We skip such proofs in what follows.
The Lie superalgebra oo IΠ (n0|n1) has a Cartan matrix if and only if n0 is odd; this matrix has the following form (up to a format; all possible formats -corresponding to * = 0 or * =0 -are described in Table  Section 14 below): If n = n0 + n1 ≥ 6, then if k0 + k1 is odd, then the Lie superalgebra oo (2) ΠΠ (n0|n1) is simple; if k0 + k1 is even, then the Lie superalgebra oo Each of these simple Lie superalgebras has a 2|4-structure; it is also close to a Lie superalgebra with Cartan matrix. To describe this CM Lie superalgebra in most simple terms, we will choose a slightly different realization of oo ΠΠ (2k0|2k1): Let us consider it as the algebra of linear transformations that preserve the bilinear form Π(2k0 + 2k1) in the format k0|k1|k0|k1. Then the algebra oo (i) ΠΠ (2k0|2k1) is spanned by supermatrices of format k0|k1|k0|k1 and the form If i ≥ 1, these derived algebras have a non-trivial central extension given by the following cocycle: (note that this expression resembles 1 2 tr(CD + C D)). We will denote this central extension of oo (i) ΠΠ (2k0|2k1) by ooc(i, 2k0|2k1). Let Then the corresponding CM Lie superalgebra is The corresponding Cartan matrix has the form (up to format; all possible formats -corresponding to * = 0 or * =0 -are described in Table Section 14 below): 6.2 The non-degenerate odd supersymmetric bilinear forms.
If m is odd, then the Lie superalgebra pe   ΠΠ is the format which in this case is m|m. Each of these simple Lie superalgebras has a 2-structure. Note that if p = 2, then the Lie superalgebra pe B (m) and its derived algebras are not close to CM Lie superalgebras (because, for example, their root system is not symmetric). If p = 2 and m ≥ 3, then they are close to CM Lie superalgebras; here we describe them.
The algebras pe Then the CM Lie superalgebras are The corresponding Cartan matrix has the form (6.7); the only condition on its format is that the last two simple roots must have distinct parities. The corresponding Dynkin diagram is shown in Table Section 14; all its nodes, except for the "horns", may be both ⊗ or , see (7.1).

Superdimensions
The following expressions (with a + sign) are the superdimensions of the relatives of the orthoorthogonal and periplectic Lie superalgebras that possess Cartan matrices. To get the superdimensions of the simple relatives, one should replace +2 and +1 by −2 and −1, respectively, in the two first lines and the four last ones: 6.3.1 Summary: The types of Lie superalgebras preserving non-degenerate symmetric forms We have the following types of non-isomorphic Lie (super) algebras (except for an occasional isomorphism intermixing the types, e.g., oo ΠΠ (6|2) pe (1) (4)): no relative has Cartan matrix with Cartan matrix The superdimensions are as follows (in the second and third column stand the additions to the superdimensions in the first column): (6.14)

Dynkin diagrams
A usual way to represent simple Lie algebras over C with integer Cartan matrices is via graphs called, in the finite dimensional case, Dynkin diagrams (DD). The Cartan matrices of certain interesting infinite dimensional simple Lie superalgebras g (even over C) can be non-symmetrizable or (for any p in the super case and for p > 0 in the non-super case) have entries belonging to the ground field K. Still, it is always possible to assign an analog of the Dynkin diagram to each (modular) Lie (super)algebra (with Cartan matrix, of course) provided the edges and nodes of the graph (DD) are rigged with an extra information. Although these analogs of the Dynkin graphs are not uniquely recovered from the Cartan matrix (and the other way round), they give a graphic presentation of the Cartan matrices and help to observe some hidden symmetries. Namely, the Dynkin diagram of a normalized n × n Cartan matrix A is a set of n nodes connected by multiple edges, perhaps endowed with an arrow, according to the usual rules [28] or their modification, most naturally formulated by Serganova: compare [47,20] with [21]. In what follows, we recall these rules, and further improve them to fit the modular case.

Digression
Let ξ = (ξ 1 , . . . , ξ n ) and η = (η 1 , . . . , η n ) be odd elements, p = (p 1 , . . . , p m ), q = (q 1 , . . . , q m ) and z even elements. hei(2m|2n) = Span(p, q, ξ, η, z), where the brackets are In what follows we will need the Lie superalgebra hei(2m|2n) (for the cases where mn = 0) and its only (up to the change of parity) non-trivial irreducible representation, called the Fock space, which in characteristic p is K[q, ξ]/(q p 1 , . . . , q p n ) on which the elements q i and ξ j act as operators of left multiplication by q i and ξ j , respectively, whereas p i and η j act as h∂ q i and h∂ ξ j , where h ∈ K \ {0} can be fixed to be equal to 1 by a change of the basis.

Edges
If p = 2 and dim g(A) < ∞, the Cartan matrices considered are symmetric. If A ij = a, where a = 0 or 1, then we rig the edge connecting the ith and jth nodes by a label a.
If p > 2 and dim g(A) < ∞, then A is symmetrizable, so let us symmetrize it, i.e., consider DA for an invertible diagonal matrix D. Then, if (DA) ij = a, where a = 0 or −1, we rig the edge connecting the ith and jth nodes by a label a.
If all off-diagonal entries of A belong to Z/p and their representatives are selected to be non-positive integers, we can draw the DD as for p = 0, i.e., connect the ith node with the jth one by max(|A ij |, |A ji |) edges rigged with an arrow > pointing from the ith node to the jth if |A ij | > |A ji | or in the opposite direction if |A ij | < |A ji |.

Ref lections
Let R + be a system of positive roots of Lie superalgebra g, and let B = {σ 1 , . . . , σ n } be the corresponding system of simple roots with some corresponding pair (A = A B , I = I B ). Then for any k ∈ {1, . . . , n}, the set (R + \{σ k }) {−σ k } is a system of positive roots. This operation is called the reflection in σ k ; it changes the system of simple roots by the formulas where we consider Z/pZ as a subfield of K.
7.3.1. Remark. In the second, fourth and penultimate cases, the matrix entries in (7.4) can, in principle, be equal to kp − 1 for any k ∈ N, and in the last case any element of K may occur. We may only hope at this stage that, at least for dim g < ∞, this does not happen.
The values − 2A kj A kk and − A kj A kk are elements of K, while the roots are elements of a vector space over R. Therefore These expressions in the first and third cases in (7.4) should be understood as these expressions are always congruent to integers.) There is known just one exception: If p = 2 and A kk = A jk , the expression − 2A jk A kk should be understood as 2, not 0. The name "reflection" is used because in the case of (semi)simple finite-dimensional Lie algebras this action extended on the whole R by linearity is a map from R to R, and it does not depend on R + , only on σ k . This map is usually denoted by r σ k or just r k . The map r σ i extended to the R-span of R is reflection in the hyperplane orthogonal to σ i relative the bilinear form dual to the Killing form.
The reflections in the even (odd) roots are referred to as even (odd) reflections. A simple root is called isotropic, if the corresponding row of the Cartan matrix has zero on the diagonal, and non-isotropic otherwise. The reflections that correspond to isotropic or non-isotropic roots will be referred to accordingly.
If there are isotropic simple roots, the reflections r α do not, as a rule, generate a version of the Weyl group because the product of two reflections in nodes not connected by one (perhaps, multiple) edge is not defined. These reflections just connect pair of "neighboring" systems of simple roots and there is no reason to expect that we can multiply two distinct such reflections. In the general case (of Lie superalgebras and p > 0), the action of a given isotropic reflections (7.3) can not, generally, be extended to a linear map R −→ R. For Lie superalgebras over C, one can extend the action of reflections by linearity to the root lattice but this extension preserves the root system only for sl(m|n) and osp(2m + 1|2n), cf. [48].
If σ i is an odd isotropic root, then the corresponding reflection sends one set of Chevalley generators into a new one:

A careful study of an example
Now let p = 2 and let us apply all the above to the Lie superalgebra pe(k) (the situation with o Π (2k) and oo ΠΠ (2k0|2k1) is the same). For the Cartan matrix (all possible formatscorresponding to * = 0 or * =0 -are listed in Table Section 14) we take The Lie superalgebra pe (i) (k) consists of supermatrices of the form where for i = 0, we have B ∈ gl(k), C, D are symmetric; for i = 1, we have B ∈ gl(k), C, D are symmetric zero-diagonal; for i = 2, we have B ∈ sl(k), C, D are symmetric zero-diagonal.
We expect (by analogy with the orthogonal Lie algebras in characteristic = 2) that Let us first consider the (simpler) case of k odd. Then rk A = k − 1 since the sum of the last two rows is zero. Let us start with the simple algebra pe (2) .3) are what we expect them to be from their p = 0 analogs. But from the definition of CM Lie superalgebra we see that the algebra must have a center z, see (8.4). Thus, the CM Lie superalgebra is not pe (2) (k) but is spanned by the central extension of pe (2) (k) plus the grading operator defined from (4.6). The extension pec(2, k) described in (  The matrix I 0 = diag(1 k , 0 k ) satisfies all these conditions. Thus, the corresponding CM Lie superalgebra is Remark. Rather often we need ideals of CM Lie (super)algebras that do not contain the outer grading operator(s), cf. Section 4.1. These ideals, such as pec(2, k) or sl(n|n), do not have Cartan matrix.
Now let us consider the case of k even. Then the simple algebra is pe (2) (k)/(K1 2k ). The Cartan matrix is of rank k − 2: (a) the sum of the last two rows is zero; (b) the sum of all the rows with odd numbers is zero. (8.7) The condition (8.7a) gives us the same central extension and the same grading operator an in the previous case.

Main steps of our classif ication
In this section we deal with Lie (super)algebras of the form g(A) or their simple subquotients g(A) (i) /c, where i = 1 or 2.

Step 1: An overview of known results
Lie algebras (nothing super). There are known the two methods of classification: 1) Over C, Cartan [10] did not use any roots, instead he used what is nowadays called in his honor Cartan prolongations and a generalization (which he never formulated explicitly) of this procedure which we call CTS-ing (Cartan-Tanaka-Shchepochkina prolonging).
2) Nowadays, to get the shortest classification of the simple finite dimensional Lie algebras, everybody (e.g. [8,41]) uses root technique and the non-degenerate invariant symmetric bilinear form (the Killing form).
In the modular case, as well as in the super case, and in the mixture of these cases we consider here, the Killing form might be identically zero. However, if the Cartan matrix A is symmetrizable (and indecomposable), on the Lie (super)algebra g is not simple), there is a non-degenerate replacement of the Killing form. (Astonishingly, this replacement might sometimes be not coming from any representation, see [46]. Much earlier Kaplansky observed a similar phenomenon in the modular case and associated the non-degenerate bilinear form with a projective representation. Kaplansky pointed at this phenomenon in his wonderful preprints [31] which he modestly did not publish.) In the modular case, and in the super case for p = 0, this approach -to use a non-degenerate even invariant symmetric form in order to classify the simple algebras -was pursued by Kaplansky [31].
For p > 0, Weisfeiler and Kac [54] gave a classification, but although the idea of their proof is OK, the paper has several gaps and vague notions (the Brown algebra br(3) was missed, whereas Brown [9] who discovered it did not write that it possesses Cartan matrix, actually two inequivalent matrices first observed by Skryabin [50,30]; the notion of the Lie algebra with Cartan matrix nicely formulated in [28] was not properly developed at the time [54] was written; the Dynkin diagrams mentioned there were not defined at all in the modular case; the algebras g(A) and g(A) (i) /c were sometimes identified). The case p > 3 being completely investigated by Block, Wilson, Premet and Strade [42,52] (see also [1]), we double-checked the cases where p < 5. The answer of [54]∪ [50] is correct.
Lie superalgebras. Over C, for any Lie algebra g0, Kac [29] listed all g0-modules g1 such that the Lie superalgebra g = g0 ⊕ g1 is simple. (9.1) Kaplansky [31,22,32], Djoković and Hochschild [15], and also Scheunert, Nahm and Rittenberg [45] had their own approaches to the problem (9.1) and solved it without gaps for various particular cases, but they did not investigate which of the simple finite dimensional Lie superalgebras possess Cartan matrix.
Kac observed that (a) some of the simple Lie superalgebras (9.1) possess analogs of Cartan matrix, (b) one Lie superalgebra may have several inequivalent Cartan matrices. His first list of inequivalent Cartan matrices (in other words, distinct Z-gradings) for finite dimensional Lie superalgebras g(A) in [29] had gaps; Serganova [47] and (by a different method and only for symmetrizable matrices) van de Leur [53] fixed the gaps and even classified Lie superalgebras of polynomial growth (for the proof in the non-symmetrizable case, announced 20 years earlier, see [26]). Kac also suggested analogs of Dynkin diagrams to graphically encode the Cartan matrices.
Kaplansky was the first (see his newsletters in [31]) to discover the exceptional algebras ag(2) and ab(3) (he dubbed them Γ 2 and Γ 3 , respectively) and a parametric family osp(4|2; α) (he dubbed it Γ(A, B, C))); our notations reflect the fact that ag(2)0 = sl(2) ⊕ g(2) and ab(3)0 = sl(2) ⊕ o(7) (o(7) is B 3 in Cartan's nomenclature). Kaplansky's description (irrelevant to us at the moment except for the fact that A, B and C are on equal footing) of what we now identify as osp(4|2; α), a parametric family of deforms of osp(4|2), made an S 3 -symmetry of the parameter manifest (to A. A. Kirillov, and he informed us, in 1976). Indeed, since A + B + C = 0, and α ∈ C ∪ ∞ is the ratio of the two remaining parameters, we get an S 3 -action on the plane A + B + C = 0 which in terms of α is generated by the transformations: This symmetry should have immediately sprang to mind since osp(4|2; α) is strikingly similar to wk(3; a) found 5 years earlier, cf. (9.5), and since S 3 SL(2; Z/2). The following figure depicts the fundamental domains of the S 3 -action. The other transformations generated by (9.2) are The number of the matrix A such that g(A) has only one odd simple root is boxed , that with all simple roots odd is underlined. The nodes are numbered by small boxed numbers; the curly lines with arrows depict odd reflections.
1) 6 To the incredulous reader: The Cartan subalgebra of sp(4) is generated by h2 and 2h1 + h2.  1). We encounter several more instances of non-fundamental weights in descriptions of exceptions for p = 2.

9.2
Step 2: Studying 2 × 2 and 3 × 3 Cartan matrices 1) We ask Mathematica to construct all possible matrices of a specific size. The matrices are not normalized and they must not be symmetrizable: we can not eliminate non-symmetrizable matrices at this stage. Fortunately, all 2 × 2 matrices are symmetrizable.
2) We ask Mathematica to eliminate the matrices with the following properties: a) Matrices A for whose submatrix B we know that dim g(B) = ∞; b) decomposable matrices. (9.12) 3) Matrices with a row in each that differ from each other by a nonzero factor are counted once, e.g., 1 1 3 2 ∼ = 2 2 3 2 ∼ = 6 6 6 4 .

4)
Equivalent matrices are counted once, where equivalence means that one matrix can be obtained from the other one by simultaneous transposition of rows and columns with the same numbers and the same parity. For example, At the substeps 1.1)-1.4) we thus get a store of Cartan matrices to be tested further.
5) Now, we ask SuperLie, see [25], to construct the Lie superalgebras g(A) up to certain dimension (say, 256). Having stored the Lie superalgebras g(A) of dimension < 256 we increase the range again if there are any algebras left (say, to 1024 or 2048). At this step, we conjecture that the dimension of any finite dimensional simple Lie (super)algebra of the form g(A), where A is of size n × n, does not grow too rapidly with n. Say, at least, not as fast as n 10 .
If the dimension of g(A) increases accordingly, then we conjecture that g(A) is infinite dimensional and this Lie superalgebra is put away for a while (but not completely eliminated as decomposable matrices that correspond to non-simple algebras: The progress of science might require soon to investigate how fast the dimension grows with n: polynomially or faster). 6) For the stored Cartan matrices A, we have dim g(A) < ∞. Once we get the full list all of such Cartan matrices of a given size, we have to check if g(A) is simple, one by one.

The case of 2 × 2 Cartan matrices
On the diagonal we may have 2,1 or0, if the corresponding root is even; 0 or 1 if the root is odd. To be on the safe side, we redid the purely even case. We have the following options to consider:  Obviously, some of these CMs had appeared in the study of (twisted) loops and the corresponding Kac-Moody Lie (super)algebras. One could expect that the reduction of the entries of A modulo p might yield a finite dimensional algebra, but this does not happen.
Proof . We prove this by inspection for 3 × 3 matrices, but the general case does not follow by reduction and induction: For example, for p = 2 and the normalized non-symmetrizable matrix (here the value of * is irrelevant) where a = 0, 1, or analogous n × n matrix whose Dynkin diagram is a loop, any 3 × 3 submatrix is symmetrizable.
To eliminate non-symmetrizable Cartan matrices, and any loops of length > 3 in Dynkin diagrams, is, nevertheless, possible using Lemmas 3.1, 3.3, and 3.10, 3.11 of [54]. (Van de Leur [53] used these Lemmas for p = 0.) That was the idea of the proof. Now we pass to the case-by-case study.

9.3
Step 3: Studying n × n Cartan matrices for n > 3 By Lemma 9.2.2 we will assume that A is symmetrizable. The idea is to use induction and the information found at each step. 9.3.1. Hypothesis. Each finite dimensional Lie superalgebra of the form g(A) possesses a "simplest" Dynkin diagram -the one with only one odd node.
Therefore passing from n × n Cartan matrices to (n + 1) × (n + 1) Cartan matrices it suffices to consider just two types of n × n Cartan matrices: Purely even ones and the "simplest" oneswith only one odd node on their Dynkin diagrams. To the latter ones only even node should be added.

Further simplif ication of the algorithm
Enlarging Cartan matrices by adding new row and column, we let, for n > 4, its only non-zero elements occupy at most four slots (apart from the diagonal). Justification: Lemmas from § 3 in [54] and Lemma 9.2.2.
Even this simplification still leaves lots of cases: To the 5 cases to be enlarged for Cartan matrices of size ≤ 8 that we encounter for p = 0, we have to add 16 super cases, each producing tens of possibilities in each of the major cases p = 2, 3 and 5. To save several pages per each n for each p, we have omitted the results of enlargements of each Cartan matrix and give only the final summary.

On a quest for parametric families
Even for 2 × 2 Cartan matrices we could have proceeded by "enlarging" but to be on the safe side we performed the selection independently. We considered only one or two parameters using the function called ParamSolve (of SuperLie, see [25]). It shows all cases where the division by an expression possibly equal to zero occurred. Every time SuperLie shows such a possibility we check it by hand; these possibility are algebraic equations of the form β = f (α), where α and β are the parameters of the CM. We saw that whenever α and β are generic dim g(A) grows too fast as compared with the height of the element (i.e., the number of brackets in expressions like [a, [b, [c, d]]]) that SuperLie should not exceed constructing a Lie (super)algebra. We did not investigate if the growth is polynomial or exponential, but definitely dim g(A) = ∞. For each pair of singular values of parameters β = f (α), we repeat the computations again. In most cases, the algebra is infinite-dimensional, the exceptions being β = α + 1 that nicely correspond to some of CMs we already know, like wk algebras.
For three parameters, we have equations of the form γ = f (α, β). For generic α and β. the Lie superalgebra g(A) is infinite-dimensional. For the singular cases given by SuperLie, the constraints are of the form β = g(α). Now we face two possibilities: If γ is a constant, then we just use the result of the previous step, when we dealt with two parameters. In the rare cases where γ is not a constant and depends on the parameter α, we have to recompute again and again the dim g(A) is infinite in these cases.
We find Cartan matrices of size 4 × 4 and larger by "enlarging". For p = 2, we see that 3 × 3 CMs with parameters can be extended to 4 × 4 CMs. However, 4 × 4 CMs cannot be extended to 5 × 5 CMs whose Lie (super)algebras are of finite dimension. For p > 2, even 3 × 3 CMs cannot be extended.

Super and modular cases: Summary of new features
(as compared with simple Lie algebras over C) The super case, p = 0.
1) There are three types of nodes ( • , ⊗ and • ), 2) there may occur a loop but only of length 3; 3) there is at most 1 parameter, but 1 parameter may occur; 4) to one algebra several inequivalent Cartan matrices can correspond.
The modular case. For Lie algebras, new features are the same as in the p = 0 super case; additionally there appear new types of nodes ( and * ). 10 The answer: The case where p > 5 This case is the simplest one since it does not differ much from the p = 0 case, where the answer is known.

11
The answer: The case where p = 5 Simple Lie algebras: 1) same as in Section 10 for p = 5.

12
The answer: The case where p = 3 Simple Lie algebras: 1) same as in Section 10 for p = 3, except g(2) which is not simple but contains a unique minimal ideal isomorphic to psl(3), and the following additional exceptions: 2) the Brown algebras br(2; a) and br(2) as well as br (3), see Section 9.1.

Elduque and Cunha superalgebras: Systems of simple roots
For details of description of Elduque and Cunha superalgebras in terms of symmetric composition algebras, see [17,13,14]. Here we consider the simple Elduque and Cunha superalgebras with Cartan matrix for p = 3. In what follows, we list them using somewhat shorter notations as compared with the original ones: Hereafter g(A, B) denotes the superalgebra occupying (A, B)th slot in the Elduque Supermagic Square; the first Cartan matrix is usually the one given in [13], where only one Cartan matrix is given; the other matrices are obtained from the first one by means of reflections. Accordingly, ig(A, B) is the shorthand for the realization of g(A, B) by means of the ith Cartan matrix. There are no instances of isotropic even reflections. On notation in the following tables, see Section 9.1.1.

Simple Lie superalgebras
In the list below the term "super version" of a Lie algebra g(A) stands for a Lie superalgebra with the "same" root system as that of g(A) but with some of the simple roots considered odd.
1) The Lie superalgebras obtained from their p = 0 analogs that have no −2 in off-diagonal slots of the Cartan matrix by reducing the structure constants modulo 2 (for osp(2n+1|2m) one has, first of all, to divide the last row by 2 in order to normalize CM), we thus get the CM versions of sl, namely: either simple sl(a|a + 2k + 1) or gl(a|2k + a) whose "simple core" is psl(a|a + 2k) for a odd or psl(a|a + 2k) (1) for a even; 2) the ortho-orthogonal algebras, namely: oo (1) and ooc whose "simple cores" are described in Section 6; 13.1 On the structure of bgl(3; α), bgl(4; α), and e(a, b) In this section we describe the even parts g0 of the new Lie superalgebras g = g(A) and their odd parts g1 as g0-modules. SuperLie enumerates the elements of the Chevalley basis the x i (positive), starting with the generators, then their brackets, etc., and the y i are negative root vectors opposite to the x i . Since the irreducible representations of the Lie algebras may have neither highest nor lowest weight, observe that the g0-modules g1 always have both highest and lowest weights. In these four cases, g(A)0 is of the form where the matrix B is not invertible (so g(B) has a grading element d and a central element c), and where X + , X − and c span the Heisenberg Lie algebra hei (2). The brackets are: The odd part of g(A) (at least in two of the four cases) consists of two copies of the same g(B)module N , the operators ad X ± permute these copies, and ad 2 X ± = 0, so each of the operators maps one of the copies to the other, and this other copy to zero.

The e-type superalgebras
Notation. The e-type superalgebras will be denoted by (one of) their simplest Dynkin diagrams, i.e., e(n, i) denotes the Lie superalgebra whose diagram is of the same shape as that of the Lie algebra e(n) but with the only -ith -node ⊗. This, and other "simplest", Cartan matrices are boxed. We enumerate the nodes of the Dynkin diagram of e(n) as in [8,41]: We first enumerate the nodes in the row corresponding to sl(n) (from the end-point of the "longest" twig towards the branch point and further on along the second long twig), and the nth node is the end-point of the shortest "twig". e(6, 1) e(6, 5) of sdim = 46|32. We have g0 oc(2; 10)⊕KZ and g1 is a reducible module of the form R(π 4 ) ⊕ R(π 5 ) with the two highest weight vectors and y 5 . Denote the basis elements of the Cartan subalgebra by Z, h 1 , h 2 , h 3 , h 4 , h 6 . The weights of x 36 and y 5 are respectively, (0, 0, 0, 0, 0, 1) and (0, 0, 0, 0, 1, 0). The module generated by x 36 gives all odd positive roots and the module generated by y 5 gives all odd negative roots. e(6, 6) of sdim = 38|40. In this case, g(B) gl (6), see Section 13.1.1. The module g1 is irreducible with the highest weight vector ]]] of weight (0, 0, 1, 0, 0, 1).
The module g1 is irreducible with the highest weight vector The Cartan subalgebra is spanned by h 1 + h 3 + h 5 , h 1 , h 2 , h 3 , h 4 , h 7 and also h 6 and d 1 . The weight of x 62 is (1, 0, 0, 0, 0, 0, 1, 0). The highest weight vector of g1 is the highest weight vector of one of the copies of the g(B)-module N , see Section 13.1.1, so the highest weight of N is the same as the highest weight of g1. (Of course, this is true for the other two similar cases as well; in the case of e(6, 6), we used Lebedev's choice -another basis of h -and expressed the weight with respect to it.) e(7, 7) of sdim = 64/62|70. Since the Cartan matrix of this Lie superalgebra is of rank 6, a grading operator d 1 should be (and is) added. Then g0 gl (8). The module g1 has the two highest weight vectors: and y 7 . The Cartan subalgebra is spanned by h 1 , h 2 , h 3 , h 4 , h 5 , h 6 and also h 1 + h 3 + h 7 and d 1 .
13.2 Systems of simple roots of the e-type Lie superalgebras 13.2.1. Remark. Observe that if p = 2 and the Cartan matrix has no parameters, the reflections do not change the shape of the Dynkin diagram. Therefore, for the e-superalgebras, it suffices to list distributions of parities of the nodes in order to describe the Dynkin diagrams. Since there are tens and even hundreds of diagrams in these cases, this possibility saves a lot of space, see the lists of all inequivalent Cartan matrices of the e-type Lie superalgebras.
13.2.2 e(6, 1) e(6, 5) of sdim 46|32 All inequivalent Cartan matrices are as follows (none of the matrices corresponding to the symmetric pairs of Dynkin diagrams is excluded but are placed one under the other for clarity, followed by three symmetric diagrams): . . .
if m is even. m

Notation
The Dynkin diagrams in Table 14 correspond to CM Lie superalgebras close to ortho-orthogonal and periplectic Lie superalgebras. Each thin black dot may be ⊗ or ; the last five columns show conditions on the diagrams; in the last four columns, it suffices to satisfy conditions in any one row. Horizontal lines in the last four columns separate the cases corresponding to different Dynkin diagrams. The notations are: v is the total number of nodes in the diagram; ng is the number of "grey" nodes ⊗'s among the thin black dots; png is the parity of this number; ev and od are the number of thin black dots such that the number of ⊗'s to the left from them is even and odd, respectively.
15 Fixed points of symmetries of the Dynkin diagrams

Recapitulation
For p = 0, it is well known that the Lie algebras of series B and C and the exceptions F and G are obtained as the sets of fixed points of the outer automorphism of an appropriate Lie algebra of ADE series. All these automorphisms correspond to the symmetries of the respective Dynkin diagram. Not all simple finite dimensional Lie superalgebras can be obtained as the sets of fixed points of the symmetry of an appropriate Dynkin diagram, but many of them can, see [21]. Recall Serganova's result [47] on outer automorphisms (i.e., the modulo the connected component of the unity of the automorphism group) of simple finite dimensional Lie superalgebras for p = 0. The symmetry of the Dynkin diagram of sl(n) corresponds to the transposition with respect to the side diagonal, conjugate in the group of automorphisms of sl(n) to the "minus transposition" X → −X t . In the super case, this automorphism becomes X −→ −X st , where This automorphism, seemingly of order 4, is actually of order 2 modulo the connected component of the unity of the automorphism group, and is of order 4 only for sl(2n + 1|2m + 1). The queer Lie superalgebra q(n) is obtained as the set of fixed points of the automorphism Π :

New result
The modular version of the above statements is given in the next Theorem in which, speaking about ortho-orthogonal and periplectic superalgebras, we distinguish the cases where the fork node is grey or white (gg(A) and wg(A), respectively); to squeeze the data in the table, we write g instead of g ⊂ + KI 0 . We also need the following decomposable Cartan matrices (p = 2):

The Lie algebra g(N )
It is of dim 34 and not simple; it contains a simple ideal of dim = 26 which is o(1; 8) (1) /c and the quotient is isomorphic to sl(3).

The Lie superalgebra g(M)
It is of sdim 18|16 and not simple. Its even part is hei(2) ⊕ c g(C), where hei(2) = Span{X ± , c} and c is the center of the Lie algebra g(C), where Its weight is (0, 0, 1, 0) (according to the ordering of the generators of the Cartan subalgebra as above).
The restriction of the module to hei(2) consists of 8 copies of the 2-dimensional irreducible Fock module; the restriction to g(C) consists of 2 copies of an irreducible 8-dimensional module.
Replacing E by E + λ · c (this does not affect the commutation relations in g0), we may assume that ad E |g1 = deg. Let us try to define the bracket on g1. I claim that it suffices to determine the only bracket (determine x) For g to be simple, we should have [g1, g1] = g0, i.e., x / ∈ g 0 . But ϕ 0 and ϕ * 0 are eigenvectors of the operator E of weight 0 and n, respectively. Hence their bracket, x, is an eigenvector of weight n and, since x should have a non-zero projection to E, the number n must be even: where α = 0.
Note that we have one more degree of freedom: We can multiply all elements of g1 by the same non-zero scalar. This helps us to fix α = 1. Thus, x = E + β · c. Now, using equations (16.4)-(16.5) we can recover the general formula for the bracket in g1. I claim that it is of the following beautiful form [f, g] = c (ad E (f )g)+ (E + βc) (ad c (f )g)+ p i (ad q i (f )g)+ q i (ad p i (f )g , (16.6) where is the Berezin integral = the coefficient of the highest term (once the basis of the Grassmann algebra is chosen). To see this, it suffices to verify that (16.6) is invariant with respect to ad q i and ad p i (the invariance with respect to ad c and ad E is obvious).
Here comes an incomplete argument. It remains to verify the Jacobi identity only for triples of odd elements, moreover, it suffices to check it only for triples of the form ϕ 0 , ϕ * 0 , ϕ. Observe that the first summand can only be non-zero if ϕ = ϕ * 0 or ϕ * i , the second summand can only be non-zero if ϕ = ϕ 0 or ϕ i .
Here comes the complete argument: In the above argument we forgot that the Jacobi identity for p = 2 and odd elements is of the form (3.3), not of the usual form [x, [x, x]] = 0. And taking x = ϕ 0 , y = p 1 we fail to satisfy the Jacobi identity although it is so tempting to set x 1 := q 1 , x 2 := ϕ 1 , y 1 := p 1 , y 2 := ϕ 2 and (correctly) deduce that the relations between these x's and y's yield the same Cartan matrix of G(4) as that of oo (1) IΠ (1|4). However, due to squaring the space of G(4) is not a Lie superalgebra. There are, however, simple Lie superalgebras with solvable even part other than oo ΠΠ (4|4), (16.11) though simple (perhaps, modulo center), have solvable even parts. One can not get a simple Lie superalgebra from oo (1) ΠΠ (2|2) passing to derived and factorizing.