Singular Degenerations of Lie Supergroups of Type $D(2,1;a)$

The complex Lie superalgebras $\mathfrak{g}$ of type $D(2,1;a)$ - also denoted by $\mathfrak{osp}(4,2;a) $ - are usually considered for"non-singular"values of the parameter $a$, for which they are simple. In this paper we introduce five suitable integral forms of $\mathfrak{g}$, that are well-defined at singular values too, giving rise to"singular specializations"that are no longer simple: this extends the family of simple objects of type $D(2,1;a)$ in five different ways. The resulting five families coincide for general values of $a$, but are different at"singular"ones: here they provide non-simple Lie superalgebras, whose structure we describe explicitly. We also perform the parallel construction for complex Lie supergroups and describe their singular specializations (or"degenerations") at singular values of $a$. Although one may work with a single complex parameter $a$, in order to stress the overall $\mathfrak{S}_3$-symmetry of the whole situation, we shall work (following Kaplansky) with a two-dimensional parameter $\boldsymbol{\sigma} = (\sigma_1,\sigma_2,\sigma_3)$ ranging in the complex affine plane $\sigma_1 + \sigma_2 + \sigma_3 = 0$.

In the classification of simple, finite dimensional Lie superalgebras over C a special oneparameter family occurs, whose elements g a depend on a parameter a ∈ C \ {0, −1} and are said to be of type D(2, 1; a) . Roughly speaking, these are "generically non-isomorphic", namely there is a group of isomorphisms Γ ∼ = S 3 freely acting on the family g a a∈C\{0,−1} .
This notation has two origins: (i) this Lie superalgebra is just osp(4, 2) when a ∈ 1, −2, − 1 2 , and (ii) this becomes a family of Lie algebras over a field of characteristic 2 , as was shown in [KV]. A drawback of this notation is that,à priori, one does not see from it the built-in S 3 -symmetry; in this respect, the notation introduced by I. Kaplansky [Kap] instead, that is Γ(A, B, C) , seems to be more reasonable; however, Kaplansky's notation also has a defect, that is one cannot guess out of it any particular property beside S 3 -symmetry. In this paper we adopt Kac' notation D(2, 1; a) since it definitely seems, nowadays, the most commonly used and known in literature. Notice also that the Cartan matrix in [KV] is essentially the same as the one we use in §3.1.1.
On top of each of the (simple) Lie superalgebras g a one can construct a corresponding Lie supergroup, say G a ; this can be done via the equivalence between super Harish-Chandra pairs and Lie supergroups (like, e.g., in [Ga3]), or also -in an algebro-geometric settingvia the construction of "Chevalley supergroups" (as in [FG] and [Ga1]). Any such G a has g a as its tangent Lie superalgebra, and overall they form a family G a a∈C\{0,−1} bearing again a S 3 -action that integrates the S 3 -action on the family g a a∈C\{0,−1} .
The starting point of the present paper is the following remark: the definition of g a , if suitable (re)formulated, still makes sense for the "singular values" a = 0 and a = −1 alike. Indeed, one can describe g a at "non-singular" values of the parameter a choosing a suitable basis -hence a corresponding integral form -and then use that same basis to define g a at singular values as well. The aim of this article is to show this dependency on the choice of integral form of the Lie superalgebra g a and its corresponding supergroup. In fact, we present five (out of many) possible ways to perform such a step, i.e. five choices of bases (hence of integral forms) that lead to different outcomes. The remarkable fact then is that in each case the new Lie superalgebras g a we find at "exceptional values" of a are non simple; in this way we extend the old family g a a∈C\{0,−1} of simple Lie superalgebras to five larger families, indexed by a ∈ C , whose elements coincide for non-singular values of a but do not for the singular ones.
Indeed, our construction is more precise, as instead of working with Lie superalgebras g a indexed by a single parameter a ∈ C \ {0, −1} -later extended to a ∈ C -we rather deal with a two-dimensional multiparameter σ ∈ V := (σ 1 , σ 2 , σ 3 ) ∈ C 3 i σ i = 0 . For each σ ∈ V we define a Lie superalgebra g(σ) via Kac' standard presentation (cf. [K]) in terms of a matrix A depending on σ : so we still use Kac' language, but sticking closer to Kaplansky-Scheunert's point of view, as in [Kap] and [Sc]. Thus we have a full family of Lie superalgebras g(σ) σ∈V , forming a bundle over V , naturally endowed with an action We call Weil superalgebra any finite-dimensional commutative C-superalgebra A such that A = C ⊕ N(A) where C is even and N(A) = N(A)0 ⊕ N(A)1 is a Z 2 -graded nilpotent ideal (the nilradical of A ). Every Weil superalgebra A is endowed with the canonical epimorphisms p A : A −−−։ C and u A : C ֒−→ A , such that p A • u A = id C . Weil superalgebras over C form a full subcategory of (salg) C , denoted by (Wsalg) or (Wsalg) C . Finally, let (Walg) C := (Wsalg) C ∩(alg) C -also denoted by (Walg) -be the category of Weil algebras (over C), i.e., the full subcategory of all totally even objects in (Wsalg) C -namely, those whose odd part is trivial. Then the functor ( )0 : (salg) −→ (alg) obviously restricts to a similar functor ( )0 : (Wsalg) −→ (Walg) given again by A → A0 .
All Lie C-superalgebras form a category, denoted by (sLie) C or just (sLie) , whose morphisms are C-linear, preserving the Z 2 -grading and the bracket. Note that if g is a Lie C-superalgebra, then its even part g0 is automatically a Lie C-algebra.
Lie superalgebras can also be described in functorial language. Indeed, let (Lie) C be the category of Lie C-algebras. Then every Lie C-superalgebra g ∈ (sLie) C defines a functor L g : (Wsalg) C −−−→ (Lie) C , A → L g (A) := A ⊗ g 0 = (A0 ⊗ g0) ⊕ (A1 ⊗ g1) Indeed, A ⊗ g is a Lie superalgebra (in a suitable, more general sense, over A ) on its own, with Lie bracket a ⊗ X , a ′ ⊗ X ′ := (−1) |X| |a ′ | a a ′ ⊗ X, X ′ ; now L g (A) is the even part of A ⊗ g , hence it is a Lie algebra on its own. 2.3. Lie supergroups. We shall now recall, in steps, the notion of complex holomorphic "Lie supergroups", as a special kind of "supermanifold".
2.3.1. Supermanifolds. By superspace we mean a pair S = |S|, O S of a topological space |S| and a sheaf of commutative superalgebras O S on it such that the stalk O S,x of O S at each point x ∈ |S| is a local superalgebra. A morphism φ : S −→ T between superspaces S and T is a pair |φ| , φ * where |φ| : |S| −→ |T | is a continuous map of topological spaces and the induced morphism φ * : O T −→ |φ| * O S of sheaves on |T | is such that φ * x (m |φ|(x) ) ⊆ m x , where m |φ| (x) and m x denote the maximal ideals in the stalks O T,|φ|(x) and O S,x respectively.
As basic model, the superspace C p|q is defined to be the topological space C p endowed with the following sheaf of commutative superalgebras: O C p|q (U) := H C p (U) ⊗ C Λ C (ξ 1 , . . . , ξ q ) for any open set U ⊆ C p , where H C p is the sheaf of holomorphic functions on C p and Λ C (ξ 1 , . . . , ξ q ) is the complex Grassmann algebra on q variables ξ 1 , . . . , ξ q of odd parity.
A (complex holomorphic) supermanifold of (super)dimension p|q is a superspace M = |M| , O M such that |M| is Hausdorff and second-countable and M is locally isomorphic to C p|q , i.e., for each between holomorphic supermanifolds is just a morphism (between them) as superspaces.
We denote the category of (complex holomorphic) supermanifolds by (hsmfd) . Let now M be a holomorphic supermanifold and U an open subset in |M| . Let I M (U) be the (nilpotent) ideal of O M (U) generated by the odd part of the latter: then O M I M defines a sheaf of purely even superalgebras over |M| , locally isomorphic to H C p . Then M rd := |M| , O M I M is a classical holomorphic manifold, called the underlying holomorphic (sub)manifold of M ; the standard projection s →s := s + I M (U) (for all s ∈ O M (U) ) at the sheaf level yields an embedding M rd −→ M , so M rd can be seen as an embedded sub(super)manifold of M . The whole construction is clearly functorial in M .
Finally, each "classical" manifold can be seen as a "supermanifold", just regarding its structure sheaf as one of superalgebras that are actually totally even, i.e. with trivial odd part. Conversely, any supermanifold enjoying the latter property is actually a "classical" manifold, nothing more. In other words, classical manifolds identify with those supermanifolds M that actually coincide with their underlying (sub)manifolds M rd .
2.3.2. Lie supergroups and the functorial approach. A group object in the category (hsmfd) is called (complex holomorphic) Lie supergroup. These objects, together with the obvious morphisms, form a subcategory among supermanifolds, denoted (Lsgrp) C .
Lie supergroups -as well as supermanifolds -can also be conveniently studied via a functorial approach that we now briefly recall (cf. [BCF] or [Ga3] for details).
Let M be a supermanifold. For every x ∈ |M| and every A ∈ (Wsalg) we set M A,x = Hom (salg) O M,x , A and M A = x∈|M | M A,x ; then we define W M : (Wsalg) −→ (set) to be the "Weil-Berezin" functor given by Overall, this provides a functor B : (hsmfd) −→ [(Wsalg), (set)] given on objects by M → W M ; we can now refine still more.
Given a finite dimensional commutative algebra A0 over C , a (complex holomorphic) A0-manifold is any manifold that is locally modelled on some open subset of some finite dimensional A0-module, so that the differential of every change of charts is an A0-module isomorphism. An A0-morphism between two A0-manifolds is any morphism whose differential is everywhere A0-linear. Gathering all A0-manifolds (for all possible A ), and suitably defining morphisms among them, one defines the category (A0hmfd) of all "A0-manifolds".
The first key point now is that each functor W M actually is valued into (A0hmfd) . Furthermore, let (Wsalg) , (A0hmfd) be the subcategory of (Wsalg) , (A0hmfd) with the same objects but whose morphisms are all natural transformations φ : The final outcome is that we have a functor S : (hsmfd) −→ (Wsalg), (A0hmfd) , given on objects by M → W M ; the key result is that this embedding is full and faithful, so that for any two supermanifolds M and N one Still relevant to us, is that the embedding S preserves products, hence also group objects. Therefore, a supermanifold M is a Lie supergroup if and only if S(M) := W M takes values in the subcategory (among A0-manifolds) of group objects -thus each W M (A) is a group.
Finally, in the functorial approach the "classical" manifolds (i.e., totally even supermanifolds) can be recovered as follows: in the previous construction one simply has to replace the words "Weil superalgebras" with "Weil algebras" everywhere. It then follows, in particular, that the Weil-Berezin functor of points W M of any holomorphic, manifold M is actually a functor from (Walg) to (A0hmfd) ; one can still see it as (the Weil-Berezin functor of points of) a supermanifold -that is totally even, though -by composing it with the natural functor ( )0 : (Wsalg) −→ (Walg) . On the other hand, given any supermanifold M , say holomorphic, the Weil-Berezin functor of points of its underlying submanifold M rd is given by Finally, it is worth stressing that the functorial point of view on supermanifolds was originally developed -by Leites, Berezin, Deligne, Molotkov, Voronov and many othersin a slightly different way. Namely, they considered functors defined, rather than on Weil superalgebras, on Grassmann (super)algebras. Actually, the two approaches are equivalent: see [BCF] for a detailed, critical analysis of the matter.
There are some advantages in restricting the focus onto Grassmann algebras. For instance, they are the sheaf of the superdomains of dimensione 0|q -i.e., "super-points". Therefore, if M is a supermanifold considered as a super-ringed space, its description via a functor defined on Grassmann algebras (only) can be really seen as the true restriction of the functor of points of M , considered as a super-ringed space. Moreover, using Grassmann algebras is consistent with the development of differential super-calculus "à la De Witt".
On the other hand, the use of Weil superalgebras has the advantage that one can use it to perform differential calculus on Weil-Berezin's functors, much in the spirit of Weil's approach to differential calculus in algebraic geometry -something one cannot achieve working with Grassmann algebras only: e.g., the tangent bundle to a supermanifold, or "super-vectors" (rather than super-points) and "super-jets", or point-supported distributions, or Weil's Transitivity Theorem, etc. Note also that some peculiar properties for Grassmann algebras are still available for Weil superalgebras: e.g., the existence of "body" and "soul", key tools in all the theory (for instance, for any Lie supergroup G this implies the existence of a semidirect product splitting of the group G(A) of A-points of G ). See [BCF] for further details. In addition, Weil-Berezin functors based on Weil superalgebras (rather than Grassmann algebras only) have been also extended to a broader class of superspaces (including supermanifolds), cf. [AHW]. So the approach via Weil superalgebras seems, in a sense, more powerful.
2.4. Super Harish-Chandra pairs and Lie supergroups. A different way to deal with Lie supergroups (or algebraic supergroups) is via the notion of "super Harish-Chandra pair", that gathers together the infinitesimal counterpart -that of Lie superalgebra -and the classical (i.e. "non-super") counterpart -that of Lie group -of the notion of Lie supergroup. We recall it shortly, referring to [Ga3] (and [Ga2]) for further details.
2.4.1. Super Harish-Chandra pairs. We call super Harish-Chandra pair -or just "sHCp" in short -any pair (G , g) such that G is a (complex holomorphic) Lie group, g a complex Lie superalgebra such that g0 = Lie(G) , and there is a (holomorphic) G-action on g by Lie superalgebra automorphisms, denoted by Ad : G −−−→ Aut(g) , such that its restriction to g0 is the adjoint action of G on Lie(G) = g0 and the differential of this action is the restriction to Lie(G) × g = g0 × g of the adjoint action of g on itself. Then a morphism (Ω, ω) : G ′ , g ′ −−→ G ′′ , g ′′ between sHCp's is given by a morphism of Lie groups Ω : G ′ −→ G ′′ and a morphism of Lie superalgebras ω : g ′ −→ g ′′ such that ω g0 = dΩ and ω • Ad g = Ad Ω + (g) • ω for all g ∈ G .
We denote the category of all super Harish-Chandra pairs by (sHCp) . The key fact now is that Lie(G) is actually valued in the category (Lie) of Lie algebras, i.e. it is a functor Lie(G) : (Wsalg) −→ (Lie) . Furthermore, there exists a Lie superalgebra g -identified with the tangent superspace to G at the unit point -such that Lie(G) = L g (cf. §2.2). Moreover, for A ∈ (Wsalg) one has Lie(G)(A) = Lie G(A) , the latter being the tangent Lie algebra of the Lie group G(A) .
On the other hand, each Lie supergroup G is a group object in the category of (holomorphic) supermanifolds: therefore, its underlying submanifold G rd is in turn a group object among (holomorphic) manifolds, i.e. it is a Lie group. More precisely, the naturality of the construction G → G rd provides a functor from Lie supergroups to (complex) Lie groups.
On top of this analysis, if G is any Lie supergroup then G rd , Lie(G) is a super Harish-Chandra pair; more precisely, we have a functor Φ : (Lsgrp) C −−→ (sHCp) given on objects by G → G rd , Lie(G) and on morphisms by φ → φ rd , Lie(φ) .
2.4.3. From sHCp's to Lie supergroups. The functor Φ : (Lsgrp) C −−→ (sHCp) has a quasi-inverse Ψ : (sHCp) −−→ (Lsgrp) C that we can describe explicitly (see [Ga3], [Ga2]). Indeed, let P := G , g be a super Harish-Chandra pair, and let B := Y i i∈I be a C-basis of g1 . For any A ∈ (Wsalg) , we define G P (A) as being the group with generators the elements of the set Γ B A := G(A) This defines the functor G P on objects, and one then defines it on morphisms as follows: for any ϕ : One proves (see [Ga3], [Ga2]) that every such G P is in fact a Lie supergroup -thought of as a special functor, i.e. identified with its associated Weil-Berezin functor. In addition, the construction P → G P is natural in P , i.e. it yields a functor Ψ : (sHCp) −−→ (Lsgrp) C ; moreover, the latter is a quasi-inverse to Φ : (Lsgrp) C −−→ (sHCp) .
3. Lie superalgebras of type D(2, 1; σ) In this section, we introduce the complex Lie superalgebras that in Kac' classification (cf. [K]) are labeled as of type D(2, 1; a) ; we do follow Kac' approach, but starting with a S 3symmetric Dynkin diagram, which makes evident the internal S 3 -symmetry of the family of all these Lie superalgebras -in fact, we recover Kaplansky-Scheunert's presentation of them (see [Kap] and [Sc]). We remark that this approach is essentially the same as starting with some Cartan matrix, where the existence of its internal S 3 -symmetry is less evident.
Then, choosing special Z-integral forms of these objects, we find the "degenerations" (i.e., singular specializations) of these integral forms at critical points of the parameter space.

Definition via Dynkin diagram.
The Lie superalgebras we are interested in depend on a parameter, which can be conveniently given by a triple σ := (σ 1 , σ 2 , σ 3 ) ∈ C * 3 ∩ σ 1 + σ 2 + σ 3 = 0 . This enters in the very definition of each Lie superalgebra g = g σ , which is given by a presentation as in [K].
3.1.1. Dynkin diagram I. For any given σ := (σ 1 , σ 2 , σ 3 ) ∈ C * 3 ∩ σ 1 + σ 2 + σ 3 = 0 we consider the Dynkin diagram SINGULAR DEGENERATIONS OF LIE SUPERGROUPS OF TYPE D(2, 1; a) To this diagram, one associates the so-called Cartan matrix given by This Cartan matrix, up to some minor detail, seems to be first appeared in [KV]. It was shown that there is a simple Lie algebra (not superalgebra!) defined over a field k of characteristic 2 associated with this Cartan matrix. Notice that it was parametrized by Π ∨ is the set of simple coroots, a basis of h , (3) Π is the set of simple roots, a basis of h * , (4) β j (H β i ) = a i,j for all 1 ≤ i, j ≤ 3 . The Lie superalgebra g = g σ is, by definition, the simple Lie superalgebra generated by H β i , X ±β j i,j=1,2,3; satisfying, at least the relations (for 1 ≤ i, j ≤ 3 ) with parity H β i =0 and X ±β i =1 for all i . We remark that the set ∆ + of positive roots has the following description: The dual h * of the Cartan subalgebra has the following description: let {ε i } i=1,2,3 ⊂ h * be an orthogonal basis normalized by the conditions (ε i , ε i ) = − 1 2 σ i ( i = 1, 2, 3 ). One can verify that (β i , β j ) = −σ k with {i, j, k} = {1, 2, 3} , where the simple roots are Remark 3.1.1. Let ∆ + 0 and ∆ + 1 be the set of even (resp. odd) positive roots. One has We set now It can be checked that, for i, j ∈ {1, 2, 3} , one has is a Lie sub-(super)algebra, with [a j , a k ] = 0 for j = k , and a i is isomorphic to sl 2 since σ i = 0 . In particular, the even part g0 of the Lie superalgebra g can be described as These formulas imply that there exists X θ ∈ g θ and X −θ ∈ g −θ such that Remark 3.1.2. By our normalization, the non-trivial actions of each a i on ✷ are given by Note that this realization was known to I. Kaplansky [Kap] and was denoted by Γ(A, B, C) for suitable A, B, C ; it has been explained in an accessible form in [Sc].
3.1.2. The Lie bracket g1 × g1 → g0 . The g0-module structure we described in the previous subsubsection inspired us to think of describing the Lie superalgebra g σ completely in terms of sl 2 (such a construction was known to M. Scheunert [Sc], as we explain below). To be precise, the only structure we are left to describe is the restriction [ , ] : g1 × g1 −→ g0 in terms of "sl 2 -language". Clearly, it is enough to record only the non-zero values of this bracket among basis elements; these are the following: Let us interpret these formulas purely in terms of sl 2 -theory.
For σ ∈ C * , we define the linear map p : One can write down this map explicitly as follows: −→ a i the above map with scalar factor σ given by −2σ i ∈ C * . To be precise, the map It can be verified that the Lie superbracket [ , ] on g1 × g1 can be expressed as Remark 3.1.3. All of the above realization of g σ in terms of sl 2 -theory actually does work for any σ ∈ C 3 ∩ σ 1 + σ 2 + σ 3 = 0 . Therefore, here and henceforth we extend our Lie superalgebra g σ to any σ ∈ C 3 ∩ σ 1 + σ 2 + σ 3 = 0 . In fact, this outcome can be achieved via a different approach, that is described in detail in [Sc], Ch. I, §1, Example 5. Indeed, the construction there starts from scratch with the (classical) Lie algebra g0 := sl ⊕3 2 and its standard action on U := ⊠ 3 i=1 ✷ i ; then one constructs a suitable g0-valued bilinear bracket P on U , that depends on σ ∈ C 3 ; finally, one gives degree0 to g0 and1 to g1 := U , and provides g := g0 ⊕ U with the bilinear bracket [ , ] g uniquely given by the Lie bracket of g0 , the g0-action on g1 := U and the bracket P on g1 . In the end, one proves that this bilinear bracket [ , ] g makes g into a Lie superalgebra if and only if the condition σ 1 + σ 2 + σ 3 = 0 is fulfilled.
The following statement is proved in [loc. cit.] again: The Lie superalgebras g σ and g σ ′ are isomorphic iff there exists τ ∈ S 3 such that σ ′ and τ.σ are proportional. Moreover, the Lie superalgebra g σ is simple iff σ ∈ (C * ) 3 .
By this reason, the case σ ∈ (C * ) 3 will be said to be "general" or "generic". In short, the isomorphism classes of our g σ 's are in bijection with the orbits of the S 3 -action in the space P Here, for the reader's convenience, we relate our Dynkin diagram with a more familiar one. This can be achieved by applying the odd reflection with respect to the root β 2 , due to V. Serganova (see [Se2]).
With respect to Π ′ , the set of positive roots is given by while the coroots can be expressed as In this subsection, we provide other bases of g σ for generic σ ; for singular values of σ instead, these new "bases" -more precisely, some slightly larger spanning sets -provide new singular degenerations. In order to achieve this, we record hereafter some formulas for the Lie brackets on elements of these spanning sets.
3.2.1. A second basis. Let σ ∈ C 3 ∩ σ 1 + σ 2 + σ 3 = 0 be generic, i.e. σ ∈ (C * ) 3 . The Lie superalgebra g σ is, by definition, the complex simple Lie superalgebra associated to the Cartan matrix A σ like in §3.1.1. Thus, letting h, ,3 be a realization of A σ (as before), our Lie superalgebra g = g σ is generated by h and {X ′ ±β } β∈Π satisfying, at least, the relations (for i, j ∈ {1, 2, 3} ) It can be checked that, for i, j ∈ {1, 2, 3} , one has is a Lie sub-(super)algebra, with a ′ j , a ′ k = 0 for j = k , isomorphic to sl 2 . In particular, the even part g0 of the Lie superalgebra g can be described as Remark 3.1.1). By this remark, it follows that an isomorphism between the odd part g1 and ✷ 1 ⊠✷ 2 ⊠✷ 3 , viewed as g0-module, is given completely by the same formula as in §3.1.1; one just has to literally replace each X ±α (in §3.1.1) with X ′ ±α . Remark 3.2.2. By our normalization, the non-trivial actions of a ′ i on ✷ i are given by Hereafter we record the non-trivial commutation relations in the new basis X ′ ±α α∈∆ + ∪ H ′ β i i=1,2,3; which might be useful for later purpose: 3.2.3. Coroots in another root basis. As in §3.1.3, we fix now a different basis of simple roots, namely i=1,2,3; is given explicitly by where the second equality makes sense only if σ 1 σ 3 = 0 . Thus, for σ 1 σ 3 = 0 , our original g = g σ can be also defined via the same Dynkin diagram as in §3.1.3.
The set of positive roots with respect to Π ′ is ∆ ′,+ = α 1 , α 2 , α 3 , α 1 + α 2 , α 2 + α 3 , α 1 + α 2 + α 3 , α 1 + 2α 2 + α 3 and the corresponding coroots, in terms of the new generators H ′ α i (for all i ), are given by Notice also that, as a consequence, we have 3.2.4. A third basis. Let again σ ∈ C 3 ∩ σ 1 + σ 2 + σ 3 = 0 be generic, i.e. σ ∈ (C * ) 3 . As a third basis for g σ we choose now a suitable mixture of the two ones considered in §3.1.1 and §3.2.1 above. Namely, let us consider ; ∪ X α α∈∆ By the previous analysis, this is yet another C-basis of g σ . In addition, the previous results also provide explicit formulas for the Lie brackets among elements of this new basis; we shall write them down explicitly -and use them -later on.
4. Integral forms & degenerations for Lie superalgebras of type D 2, 1; σ) Let l be any Lie (super)algebra over a field K , and R any subring of K . By integral form of l over R , or (integral) R-form of l , we mean by definition any Lie R-sub(super)algebra t R of l whose scalar extension to K is l itself: in other words K ⊗ R t R ∼ = l as Lie (super)algebras over K. In this subsection we introduce five particular integral forms of l = g σ , and study some remarkable specializations of them. Let ∆ := ∆ + ∪ (−∆ + ) be the root system of g σ . As a matter of notation, hereafter for any σ := (σ 1 , σ 2 , σ 3 ) ∈ C 3 we denote by Z[σ] the (unital) subring of C generated by {σ 1 , σ 2 , σ 3 } .
The reader may observe that the choice of a Z[σ]-form becomes very important when one considers a singular degeneration: one cannot speak instead of the singular degeneration, in that any degeneration actually depends not only on the specific specialization value taken by σ but also on the previously chosen Z[σ]-form. Some specific features of this phenomenon are presented in Theorems 4.1.1, 4.2.1, 4.3.1 etc. 4.1. First family: the Lie superalgebras g(σ).
. Let us consider the system of C-linear generators B g := The explicit formulas for the Lie bracket given in §3.1 show that g(σ) is a Z[σ]-subsuperalgebra of g hence also an integral Z[σ]form of the latter. Thus (4.1) defines a Lie superalgebra over Z[σ] for any possible point superalgebras indexed over the complex plane V . Moreover, taking g(σ) C := C ⊗ Z[σ] g(σ) for all σ ∈ V we find a more regular situation, in a sense that now these (extended) Lie superalgebras all share C as their common ground ring. In particular, if σ i = 0 for all i ∈ {1, 2, 3} we have g(σ) C ∼ = g σ as given in §3.1.1.
In order to formalize the description of the family g(σ) C σ∈V , we proceed as follows. Let x 1 + x 2 + x 3 be the ring of global sections of the Z-scheme associated with V . In the construction of g(σ) , formally replace x to σ (hence the x i 's to the σ i 's): this does make sense, and provides a meaningful definition of a Lie superalgebra over C[x] := C ⊗ Z Z[x] , denoted by g(x) , and then also g(x) C g(x) C as Lie C-superalgebras, through the ring isomorphism In geometrical language, all this can be formulated as follows. The Lie superalgebra g(x) , whose sheaf of sections is exactly L g C [x] . This fibre bundle can be thought of as a (total) deformation space over the base space Spec C[x] , in which every fibre can be seen as a "deformation" of any other one, and also any single fibre can be seen as a degeneration of the original Lie superalgebra g(x) C . Moreover, the fibres of g(x) C =: g C(x) . Finally, it follows from our construction that these sheaf and fibre bundle do admit an action of C * × S 3 , that on the base space Spec C[x] = V ∪ {⋆} simply fixes {⋆} and is the standard C * × S 3 -action on V .
By construction and Proposition 3.1.4, when σ 1 , σ 2 , σ 3 ∈ C * , we have that the fibre C ∼ = g σ is simple as a Lie superalgebra; instead, at each closed point of the "singular locus" 3 i=1 σ i = 0 the fibre is non-simple. This follows from direct inspection, for which we need to look at the complete multiplication table of g(σ) C .
4.1.2. Non-trivial bracket relations for g(σ) C . Resuming from formulas in §3.1, for the Lie brackets among the elements of the C-spanning set B g := H 2ε 1 , H 2ε 2 , H 2ε 3 , H θ X α α∈∆ of g(σ) C we find the following table: In particular, these explicit formulas lead to the following Theorem 4.1.1. Let σ ∈ V as above, and set a i := CX 2ε i ⊕ CH 2ε i ⊕ CX −2ε i as defined after Remark 3.1.1 for all i = 1, 2, 3 .
(1) If σ i = 0 and σ j = 0 = σ k for {i, j, k} = {1, 2, 3} , then a i g(σ) C (a Lie ideal), a i ∼ = C ⊕3 and g(σ) C is the universal central extension of psl(2|2) by a i (cf. Theorem 4.7 in [IK]); in other words, there exists a short exact sequence of Lie superalgebras parallel result also holds true when working with g(σ) over the ground ring Z[σ] .
Proof. The claim follows at once by direct inspection of the formulas in §4. . Look now at B ′ g := H ′ 2ε 1 , H ′ 2ε 2 , H ′ 2ε 3 , H ′ θ X ′ α α∈∆ , that is a second C-spanning set of g := g σ . For any σ ∈ V as before, the Z[σ]-submodule is not a free Z[σ]-module in this case -contrary to what happened with (4.1). The formulas in §3.2 prove that g ′ (σ) is also an integral Z[σ]-form of g .
Notice that the above mentioned formulas do make sense for any possible σ (such that 3 i=1 σ i = 0 ), i.e. without assuming σ = 0 . Therefore (4.2) defines a Lie superalgebra over Z[σ] for any possible σ ∈ V := σ ∈ C 3 3 i=1 σ i = 0 ; thus all these g ′ (σ)'s form a family indexed over V . Moreover, taking g ′ (σ) C := C ⊗ Z[σ] g ′ (σ) we find a more regular situation, as now the latter Lie superalgebras all share C as ground ring. In particular, we find g ′ (σ) C ∼ = g σ (as given in §3.1.1) for all possible σ ∈ V .
The family g ′ (σ) C σ∈V can be described in a formal way, similar to what we did in §4.1.1, keeping the same notation, in particular Z In the construction of g ′ (σ) , replace x with σ : this yields a definition of a Lie superalgebra over Z[x] , denoted by g ′ (x) , and also g ′ (x) g ′ (x) by scalar extension. Then definitions imply that, for any σ ∈ V , we have a Lie Z[σ]-superalgebra isomorphism One can argue similarly as in §4.1.1 to have a geometric picture of the above description: this amounts to literally replacing g(σ) with g ′ (σ) , hence we leave it to the reader.

4.2.2.
Non-trivial bracket relations for g ′ (σ) C . From the formulas in §3.2, for the Lie brackets among the elements of the C-spanning set B g ′ : C we find the following table: As a consequence, these explicit formulas yield the following 18 KENJI IOHARA , FABIO GAVARINI Theorem 4.2.1. Given σ ∈ V , consider a ′ i := C X ′ 2ε i ⊕ C H ′ 2ε i ⊕ C X ′ −2ε i as above, for all i ∈ {1, 2, 3} .
Keeping notation as before, the family g ′′ (σ) C σ∈V can be described in a formal way, taking its "version over Z[x] ", denoted by g ′′ (x) -just replacing the complex parameters (σ 1 , σ 2 , σ 3 ) =: σ with a triple of formal parameters (x 1 , x 2 , x 3 ) =: x adding to zero -and its complex-based counterpart g ′′ (x) . Then the very construction implies that, for any σ ∈ V , one has a Lie Z[σ]-superalgebra isomorphism Finally, the reader can easily provide a geometric description of the family of the g ′′ (σ) C 's, just like in §4.1.1. 4.3.2. Non-trivial bracket relations for g ′′ (σ) C . From the formulas given in the previous sections, we find for the Lie brackets among the elements of the set It follows by construction that for general values of σ one has g ′′ (σ) C ∼ = g σ -indeed, switching from either side amounts to making a change of basis, nothing more; in particular, g ′′ (σ) C is simple for all general σ . Instead, at singular values of σ one has non-simple degenerations, that are explicitly described by the following result: A parallel result also holds true when working with g ′′ (σ) over the ground ring Z[σ] .
Proof. The claim follows at once by direct inspection of the formulas in §4.3.2 above. 4.4. Degenerations from contractions: the g(σ)'s and the g ′ (σ)'s. We finish our study of remarkable integral forms of g σ by introducing some other more, that all are obtained through a general construction; when specializing these forms, one obtains again degenerations of the kind that is often referred to as "contraction".
We start with a very general construction. Let R be a (commutative, unital) ring, and let A be an "algebra" (not necessarily associative, nor unitary), in some category of Rbimodules, for some "product" · : we assume in addition that Choose now τ be a non-unit in R , and correspondingly consider in A the R-submodules Fix also a (strict) ideal I R ; then set R I := R I for the corresponding quotient ring, and use notation A τ,I := A τ I A τ ∼ = R I ⊗ R A τ , F τ,I := F τ I F τ ∼ = R I ⊗ R F and C τ,I := C τ I C τ = (τ C) (Iτ C) ∼ = R I ⊗ R C τ ∼ = R I ⊗ R (τ C) . By construction we have A τ,I ∼ = F τ,I ⊕ C τ,I as an R I -module; moreover, F τ,I · F τ,I ⊆ F τ,I , F τ,I · C τ,I ⊆ C τ,I , C τ,I · F τ,I ⊆ C τ,I , C τ,I · C τ,I ⊆ τ 2 F τ,I where the last identity comes from C τ · C τ = τ 2 (C · C) ⊆ τ 2 F = τ 2 F τ and we write τ := τ mod I ∈ R I . In particular, if τ ∈ I then C τ,I · C τ,I = {0} and we get where C τ,I bears the F τ,I -bimodule structure induced from A and is given a trivial product, so that it sits inside A τ,I as a two-sided ideal, (4.7) being a semidirect product splitting. In short, for τ ∈ I this process leads us from the initial object A , that splits into A = F ⊕ C as R-module, to the final object A τ,I = F τ,I ⋉ C τ,I , now split as a semidirect product. Following [DR], §2 -and references therein -we shall refer to this process as "contraction", and also refer to A τ,I as to a "contraction of A ".
We apply now the above contraction procedure to a couple of integral forms of g σ .
First consider the case A := g(x) , F := g(x)0 and C := g(x)1 ; here the ground ring is R := Z[x] , and we choose in it τ := x 1 x 2 x 3 ; the ideal I generated by x 1 − σ 1 , x 2 − σ 2 and x 3 − σ 3 . In this case, the "blown-up" Lie superalgebras in (4.6) reads g(x) τ = g(x)0 ⊕ τ g(x)1 , that we write also with the simpler notation g(x) := g(x) τ . Now, this provides yet another coherent sheaf of Lie superalgebras over V , whose fibre g(σ) at each non-singular (closed) point σ is again a Z[σ]-integral form of our initial complex Lie superalgebra g σ . At singular points instead, the fibres of this sheaf -i.e., the singular specializations of g(x) τ -are described by a slight variation of Theorem 4.1.1, taking into account that the Lie bracket on the odd part will be trivial, because we realize them as contractions of g(x) := g(x) τ . Similarly occurs if we work over C , i.e. we consider A := g(x) C , F := g(x) C 0 and C := g(x) C 1 with ground ring R := C[x] . Hereafter we give the exact statement on singular specializations, focusing on the complex case. As a matter of notation, we set a i := a i ( := CX 2ε i ⊕ CH 2ε i ⊕ CX −2ε i , cf. Remark 3.1.1) for all 1 = 1, 2, 3 .
Proof. The claim follows directly from Theorem 4.1.1 once we take also into account the fact that the g(σ) C 's are specializations of g(x) C , and for singular values of σ any such specialization is indeed a contraction of g(σ) C , of the form g(σ) C = g(x) C τ,I for the element τ := x 1 x 2 x 3 and the ideal I := (x i − σ i i=1,2,3; . Otherwise, we can deduce the statement directly from the explicit formulas for (linear) generators of g(x) C : indeed, the latter are easily obtained as slight modification -taking into account that odd generators must be "rescaled" by the coefficient τ := x 1 x 2 x 3 -of the similar formulas in §4.1.2.
As a second instance, we consider the case A := g ′ (x) , F := g ′ (x)0 and C := g ′ (x)1 ; the ground ring is again R := Z[x] , and again we choose in it τ := x 1 x 2 x 3 and the ideal I generated by x 1 − σ 1 , x 2 − σ 2 and x 3 − σ 3 . In this second case, we have again a "blown-up" Lie superalgebra as in (4.6), that now reads g ′ (x) τ = g ′ (x)0⊕ τ g ′ (x)1 , for which we use the simpler notation g ′ (x) := g ′ (x) τ . This gives one more coherent sheaf of Lie superalgebras over V , whose fibre g ′ (σ) at each non-singular σ ∈ V is a new Z[σ]-integral form of the complex Lie superalgebra g σ we started with. Instead, the singular specializations of g(x) τ are described by a variant of Theorem 4.2.1, taking into due account that the odd part now will have trivial Lie bracket, in that those fibres are now realized as suitable "contractions" of g ′ (x) := g ′ (x) τ . The same holds if we work on the complex field, i.e. we deal with 22 KENJI IOHARA , FABIO GAVARINI A := g ′ (x) C , F := g ′ (x) C 0 and C := g ′ (x) C 1 with ground ring R := C[x] . The exact statement on singular specializations (which is the same at any singular point in V , this time), focused on the complex case, is given below. As a matter of notation, we set now §3.2.1) for all 1 = 1, 2, 3 .
Proof. Here again, the claim follows directly from Theorem 4.2.1 together with the fact that each g ′ (σ) C is a specialization of g ′ (x) C , and for singular values of σ any such specialization is indeed a contraction of g ′ (σ) C , namely of the form g ′ (σ) C = g(x) ′ C τ,I for the element τ := x 1 x 2 x 3 and the ideal I := (x i − σ i i=1,2,3; . As alternative method, one can deduce the statement via a direct analysis of the explicit formulas for (linear) generators of g(x) ′ C , which are easily obtained as slight modification -taking into account the "rescaling" of odd generators by the coefficient τ := x 1 x 2 x 3 -of the formulas in §4.2.2. , all being indexed by the points of V . Now, our analysis shows that these five families share most of their elements, namely all those indexed by "general points" σ ∈ V \ i=1,2,3 { σ i = 0 } . On the other hand, the five families are drastically different at all points in the "singular locus" S := V ∩ i=1,2,3 { σ i = 0 } . In other words, the five sheaves L g C[x] , L g ′

C[x]
, L g C[x] and of Lie superalgebras over Spec C[x] ∼ = V ∪{⋆} ∼ = A 2 C ∪{⋆} share the same stalks on all "general" points (i.e., those outside S ), and have different stalks instead on "singular" points (i.e., those in S ). Likewise, the five fibre bundles over Spec C[x] share the same fibres on all general points and have different fibres on singular points.
The outcome is, loosely speaking, that our construction provides five different "completions" of the (more or less known) family g σ σ∈V \S of simple Lie superalgebras, by adding -in five different ways -some new non-simple extra elements on top of each point of the "singular locus" S .
Finally, recall that the original complex Lie algebras g σ of type D(2, 1; σ) were described by Scheunert (see [Sc, Ch. 1, §1, Example 5]) for any σ ∈ V , i.e. also for singular values of σ . On the general locus V \ S , Scheunert's g σ coincides with the Lie superalgebra (for the same σ) of any one of our five families above. On the singular locus instead -i.e., for any σ ∈ S -a straightforward comparison one shows that g σ coincides with g ′ (σ) C , the Lie superalgebra (over σ) of our second family. In this sense, our g ′ (σ) is a Z[σ]-form of g σ for any σ ∈ V , whilst the other four families provide us different Z[σ]-forms of g σ only on the general locus V \ S , that is a dense open subset in V .

5.
Lie supergroups of type D 2, 1; σ) : presentations and degenerations In this section, we introduce (complex) Lie supergroups of type D(2, 1; σ) , basing on the five families of Lie superalgebras introduced in §4 and following the approach of §2.4.2.
For simplicity, we formulate everything over C , but the reader may see some subtleties to discuss about the Chevalley groups over a Z[σ]-algebra. The latter had been discussed in [FG] and [Ga1] for some basis, i.e. for one particular choice of Z[σ]-integral form (though with slightly different formalism); in the present case everything works similarly, up to paying attention to the σ-dependence of the commutation relations of the Z[σ]-form one chooses (cf. §4). The details are left to the reader. 5.1. First family: the Lie supergroups G σ .
C be the complex Lie superalgebra associated with σ as in §4.1, and g0 its even part. We recall that g is spanned When σ i = 0 , the Lie algebra a i is isomorphic to sl 2 : an explicit isomorphism is realized by mapping X 2ε i → σ i e , H 2ε i → σ i h and X −2ε i → σ i f , where {e , h , f } is the standard basis sl 2 . When σ i = 0 instead, a i ∼ = C ⊕3 becomes the 3-dimensional Abelian Lie algebra.
Let us now set A i := SL 2 if σ i = 0 and A i := C × C * × C if σ i = 0 , and define G := × 3 i=1 A i -a complex Lie group such that Lie(G) = g(σ) C 0 . One sees that the adjoint action of g(σ) C 0 onto g(σ) C integrates to a Lie group action of G onto g(σ) C again, so that the pair P σ := G , g(σ) C -endowed with that action -is a super Harish-Chandra pair (cf. §2.4.1); note that its dependence on σ lies within all its constituents: the structure of G , the Lie superalgebra g(σ) C , and the action of the former onto the latter. Finally, we let G σ := G P σ be the complex Lie supergroup associated with the super Harish-Chandra pair P σ trough the category equivalence given in §2.4.3.

5.1.1.
A presentation of G σ . We shall now provide an explicit presentation by generators and relations for the supergroups G σ , i.e. for the abstract groups G σ (A) , A ∈ (Wsalg) C .
To begin with, inside each subgroup A i we consider the elements c∈C is a generating set for A i .

KENJI IOHARA , FABIO GAVARINI
The complex Lie group G + is clearly generated by i∈{1,2,3} c∈C (the h θ (c)'s might be dropped, but we prefer to add them too as generators).
In addition, when we consider G as a (totally even) supergroup and we look at it as a group-valued functor G : (Wsalg) C −→ (grps) , the abstract group G(A) of its A-pointsfor A ∈ (Wsalg) C -is generated by the set Note that here the generators do make sense -as operators in GL A ⊗ g(σ) C , but also formally -since A = C ⊕ N(A) (cf. §2.1), so each a ∈ A reads as a = c + n a for some c ∈ C and a nilpotent n a ∈ N(A) , hence exp(a X 2ε i ) = exp(c X 2ε i ) exp(n a X 2ε i ) , etc., are all well-defined.
Following the recipe in §2.4.3, in order to generate the group G σ (A) := G P σ (A) , beside the subgroup G(A) we need also all the elements of the form 1 . Therefore, we introduce notation x ±θ (η) := 1 + η X ±θ , x ±β i (η) := 1 + η X ±β i for all η ∈ A1 , i ∈ {1, 2, 3} , and we consider the set Γ1(A) := x ±θ (η) , x ±β i (η) η ∈ A1 . Now, taking into account that G(A) is generated by Γ0(A) , we can modify the set of relations given in §2.4.3 by letting g ∈ G(A) range inside the set Γ0(A) : then we can find the following full set of relations (where hereafter we freely use notation e Z := exp(Z) ): 5.1.2. Singular specializations of the supergroup(s) G σ . From the very construction of the supergroups G σ , we get that G σ is simple (as a Lie supergroup) for all σ = (σ 1 , σ 2 , σ 3 ) ∈ V such that σ i = 0 for all i ∈ {1, 2, 3} .
This follows from the presentation of G σ in §5.1.1 above, or it can be seen as a direct consequence of the relation Lie(G σ ) = g(σ) = g σ and of Proposition 3.1.4.
On the other hand, the situation is different at "singular values" of the parameter σ , as the following shows: Theorem 5.1.1. Let σ ∈ V as usual.
(1) If σ i = 0 and σ j = 0 = σ k for {i, j, k} = {1, 2, 3} , then A i G σ , A i ∼ = C×C * ×C and G σ is a central extension of PSL(2|2) by A i ; in other words, there exists a short exact sequence of Lie supergroups the center of G σ , and the quotient G σ G σ rd ∼ = C ⊕8 is Abelian; in particular, G σ is a central extension of C ⊕8 by C × C * × C ×3 , i.e. there exists a short exact sequence of Lie supergroups Proof. The claim follows directly from the presentation of G σ given in §5.1.1 above, or also from the relation Lie(G σ ) = g(σ) C along with Theorem 4.1.1.

5.2.
Second family: the Lie supergroups G ′ σ . Given σ = (σ 1 , σ 2 , σ 3 ) ∈ V , let g ′ := g ′ (σ) C be the complex Lie superalgebra associated with σ as in §4.2.1, and let g ′0 be its even part. Fix the C-basis of g0 as in §3.2, and set a ′ i : . Moreover, each Lie algebra a ′ i is isomorphic to sl 2 , an explicit isomorphism being given by , h , f } is the standard basis sl 2 . It follows that g ′0 is isomorphic to sl ⊕3 2 . For each i ∈ {1, 2, 3} , let A ′ i be a copy of SL 2 , and set G ′ := A ′ 1 × A ′ 2 × A ′ 3 . By the previous analysis, Lie(G ′ ) is isomorphic to g ′0 and the Lie G ′ -action lifts to a holomorphic G ′ -action on g ′ again: in fact, one easily sees that this action is faithful too. With this action, P ′ σ := G ′ , g ′ is a super Harish-Chandra pair (cf. §2.4.1), which overall depends on P ′ σ (although G ′ alone does not). Finally, we define G ′ σ := G P ′ σ to be the complex Lie supergroup associated with the super Harish-Chandra pair P ′ σ via the equivalence of categories given in §2.4.3.

A presentation of G ′
σ . The supergroups G ′ σ can be described in concrete terms via an explicit presentation by generators and relations of all the abstract groups G ′ σ (A) , with A ranging in (Wsalg) C . To this end, we first consider the Lie group The complex Lie group G ′ is clearly generated by the set (where the h ′ θ (c)'s might be discarded, but we prefer to keep them). Then, looking at G ′ as a (totally even) supergroup thought of as a group-valued functor G ′ : (Wsalg) C −→ (grps) , each abstract group G ′ (A) of its A-points -for A ∈ (Wsalg) C -is generated by the set and all those of the form x ′ ±θ (η) : ; as our C-basis of g ′1 ; we denote the set of all the latter by Γ . Implementing the recipe in §2.4.3, and recalling that G ′ (A) is generated by Γ ′ 0 (A) , we can now slightly modify the relations presented in §2.4.3 and consider instead the following, alternative full set of relations among the generators of G ′ σ (A) : 5.2.2. Singular specializations of the supergroup(s) G ′ σ . By construction, for the supergroups G ′ σ we have that G ′ σ is simple (as a Lie supergroup) for all σ = (σ 1 , σ 2 , σ 3 ) ∈ V such that σ i = 0 for all i ∈ {1, 2, 3} .
Indeed, this follows from the presentation of G ′ σ in §5.2.1 above, but also as a fallout of the relation Lie G ′ σ = g ′ (σ) C = g σ along with Proposition 3.1.4. The situation is different at "singular values" of the parameter σ ; the precise result is Theorem 5.2.1. Let σ ∈ V =: C ×3 ∩ σ 1 + σ 2 + σ 3 = 0 as above.
(1) If σ i = 0 and σ j = 0 = σ k for {i, j, k} = {1, 2, 3} , then letting B ′ i be the Lie subsupergroup of G ′ σ defined on every A ∈ (Wsalg) C by (2) If σ h = 0 for all h ∈ {1, 2, 3} , i.e. σ = 0 , then letting G ′ σ 1 be the Lie subsupergroup of G ′ σ defined on every A ∈ (Wsalg) C by a semidirect product of Lie supergroups. In other words, there is a split short exact sequence Proof. Like for Theorem 5.1.1, the present claim can be obtained from the presentation of G ′ σ in §5.2.1, or otherwise from the relation Lie G ′ σ = g ′ (σ) along with Theorem 4.2.1.
This follows from the presentation of G ′′ σ in §5.3.1 above, and also as a consequence of the relation Lie G ′′ σ = g ′′ (σ) C = g σ along with Proposition 3.1.4. Things change, instead, for "singular values" of the parameter σ . Before seeing it, we need to introduce some further objects of interest.
For every i ∈ {1, 2, 3} and A ∈ (Wsalg) C , define in G ′′ σ (A) the subgroups that overall -as A ranges in (Wsalg) C -define Lie subsupergroups B ′′ i and K ′′ i of G ′′ σ . The following result then tells us how G ′′ σ looks like in the singular cases: Theorem 5.3.1. Given σ ∈ V , let G σ and its Lie subsupergroups B ′′ i and K ′′ i be as above.
(1) If σ i = 0 and σ j = 0 = σ k for {i, j, k} = {1, 2, 3} , then there exists a split short exact sequence , and a second short exact sequence (2) If σ h = 0 for all h ∈ {1, 2, 3} , i.e. σ = 0 , then there exists a first short exact sequence and a second short exact sequence is isomorphic to T 3 C -the (totally even) 3-dimensional complex torus. Proof. Like for Theorem 5.1.1, one can deduce the claim from the presentation of G ′′ σ in §5.3.1, or also from the relation Lie G ′′ σ = g ′′ (σ) along with Theorem 4.3.1.

5.4.
Lie supergroups from contractions: the family of the G σ 's.
Recalling the construction of G σ in §5.1, for each i ∈ {1, 2, 3} we set A i := A i (isomorphic to either SL 2 or C × C * × C depending on σ i = 0 or σ i = 0 ) and G := × 3 i=1 A i = G , a complex Lie group such that Lie G = g(σ) C 0 . Just like in §5.1, the adjoint action of g(σ) C 0 onto g(σ) C integrates to a Lie group action of G onto g(σ) C : endowed with this action, the pair P σ := G , g(σ) C is a super Harish-Chandra pair (cf. §2.4.1), by construction. Eventually, we can define to be the complex Lie supergroup associated with P σ following §2.4.3.

5.4.1.
A presentation of G σ . To describe the supergroups G σ , we provide hereafter an explicit presentation by generators and relations of all the abstract groups G σ (A) , with A ∈ (Wsalg) C . To begin with, let exp : g0 ∼ = Lie G −−−−→ G be the exponential map. Like we did in §5.1.1 for the supergroup G σ , inside each subgroup A i we consider c∈C is a generating set for A i ; also, we consider elements h θ (c) := exp c H θ for all c ∈ C . It follows that the complex Lie group G = A 1 × A 2 × A 3 is generated by c∈C (we could drop the h θ (c)'s, but we prefer to keep them among the generators).
When we think of G as a (totally even) supergroup, looking at it as a group-valued functor G : (Wsalg) C −→ (grps) , the abstract group G(A) of its A-points -for A ∈ (Wsalg) Cis generated by the set In fact, we would better stress that, by construction (cf. §5.1), we have an obvious isomorphism G ∼ = G (see §5.1.1 for the definition of G) as complex Lie groups.
This also follows from the presentation of G σ in §5.4.1 above, or as a direct consequence of the relation Lie G σ = g(σ) = g σ and of the fact that g σ ∼ = g σ when σ i = 0 for all i .
(1) We have G σ ∼ = G σ rd ⋉ G σ 1 where G σ rd ∼ = A 1 × A 2 × A 3 and G σ 1 is the supersubgroup of G σ generated by the x ±θ 's and the x ±β i 's (for all i ).
(2) If σ i = 0 then A i G σ is a central Lie subgroup in G σ with A i ∼ = C × C * × C , otherwise A i ∼ = SL 2 .
Then exp : gl g ′ (σ) C −→ GL g ′ (σ) C yields a Lie subgroup G ′ := exp g ′ (σ) C 0 in GL g ′ (σ) C which faithfully acts onto g ′ (σ) C and is such that Lie G ′ = g ′ (σ) C 0 . The pair P ′ σ := G ′ , g ′ (σ) C with this action then is a super Harish-Chandra pair (cf. §2.4.1). As alternative method, we might also construct the super Harish-Chandra pair P ′ σ via the same procedure, up to the obvious, minimal changes, adopted for P ′ σ in §5.2; indeed, one can also do the converse, namely use the present method to construct P ′ σ as well. Once we have the super Harish-Chandra pair P ′ σ , it makes sense to define G ′ σ := G P ′ σ that is the complex Lie supergroup associated with P ′ σ after the recipe in §2.4.3.
5.5.1. A presentation of G ′ σ . We shall presently describe the supergroups G ′ σ by means of an explicit presentation by generators and relations of all the abstract groups G ′ σ (A) , for all A ∈ (Wsalg) C . To begin with, let exp : g ′ 0 ∼ = Lie G ′ −−−→ G ′ be the exponential map.
Just like for the supergroup G σ in §5.1.1, inside each subgroup A ′ i we consider c∈C is a generating set for A ′ i = A ′ i ; also, we consider elements h ′ θ (c) := exp c H ′ θ for all c ∈ C . It follows that the complex Lie group G ′ = A ′ 1 × A ′ 2 × A ′ 3 is generated by i∈{1,2,3} c∈C (as before, we could drop the h ′ θ (c)'s, but we prefer to keep them among the generators). When thinking of G ′ as a (totally even) supergroup, considered as a group-valued functor G ′ : (Wsalg) C −→ (grps) , the abstract group G ′ (A) of its A-points -for A ∈ (Wsalg) C -is generated by the set i∈{1,2,3} a∈A0 (5.5) Indeed, we can also stress that, by construction (cf. §5.1), there exists an obvious isomorphism G ′ ∼ = G ′ as complex Lie groups. Now, to generate the group G ′ σ (A) := G P ′ σ (A) following the recipe in §2.4.3, we fix in g ′ (σ) C 1 the C-basis Y i i∈I = X ′ β := τ X ′ β β ∈ ∆1 = ± θ, ±β 1 , ±β 2 , ±β 3 . Then, beside the generating elements from G ′ (A) we take as generators also those of the set Γ ′ 1 (A) := x ′ ±θ (η) := 1 + η X ′ ±θ , x ′ ±β i (η) := 1 + η X ′ ±β i i∈{1,2,3} η∈A1 In geometrical terms, each family forms a fibre space, say respectively, over the base space Spec C[x] ∼ = V ∪ {⋆} ∼ = A 2 C ∪ {⋆} , whose fibres are Lie supergroups. Our result show that the fibres in the two fibre spaces do coincide at general points -where they are simple Lie supergroups -and do differ instead at singular points -where they are non-simple indeed.
As an outcome, loosely speaking we can say that our construction provides five different "completions" of the family G σ σ∈V \S of simple Lie supergroups, by adding -in five different ways (yet many others more can be made up) -some new non-simple extra elements.