1 Algebraic supergroups of Cartan type: Existence
All notation and terminology throughout this note will be as in . In particular, every supergroup we shall consider will be fine, which means that its tangent Lie algebra functor is of the form (where A ranges among all commutative superalgebras) for some Lie superalgebra over , with the additional requirement that (as -module) be free of finite rank.
The main result in  was the construction of the “Chevalley supergroups” of Cartan type, denoted as their construction depends on some suitable -module V: these are connected algebraic -supergroups, defined over , such that the complexification of their tangent Lie superalgebra be (finite dimensional) simple of Cartan type. In particular, this proves that supergroups with such properties do exist. However, the presentation in  might be obscure on this point, since the construction of is based upon the choice of V and of a suitable lattice inside it, and the existence of such data might be unclear. This point deserves to be made clear, which is what I am doing in this section.
Let be a complex Lie superalgebra which is simple of Cartan type; let then and be its root lattice and (integral) weight lattice, respectively. As explained in [5, Section 4.5], for any Chevalley supergroup associated with and a suitable -module V its group of characters is a lattice that lies between and ; indeed, it is the lattice of weights spanned by the weights (for the action of a Cartan subalgebra of ) of V itself. In addition, this V has to be finite dimensional, faithful and rational, and also has to contain an admissible lattice, say M. Thus what we need to show is the following:
For any choice of an intermediate lattice Λ lying between and , there exists a finite-dimensional, rational, faithful -module which contains an admissible lattice M and whose set of weights spans Λ.
We start recalling that the even part of is of the form where is a reductive Lie subalgebra and is a nilpotent ideal (cf. [5, Section 2.3]).
First, by classical theory of reductive Lie algebras, for any Λ as before there is a faithful, finite-dimensional, rational -module W whose weights span Λ; moreover, such a W contains a lattice N that is “admissible”, which means stable for the Kostant form of – cf. [1, Chapter VIII, Section 12.7] (taking into account that is always simple but when is of type , for then it has a one-dimensional center, and [1, Chapter VIII, Section 12.7, Theorem 2] applies again). Since is an ideal in and , the same W is also a -module (by scalar extension), the lattice N being again “admissible”, i.e. stable for ; of course W is still rational and finite dimensional.
Second, consider : this is a -module which is still finite dimensional and rational; moreover, it contains the admissible lattice , and by construction the lattice spanned by the weights of V is exactly Λ (since we initially assumed that , i.e. Λ contains all roots of ). Finally, as is simple and its action on V is non-trivial, V itself is also faithful, as required. ∎
(a) Every Cartan type simple Lie superalgebra can be realized (or defined, if you wish) as a Lie subsuperalgebra of some for a suitable V – more precisely, as a suitable Lie superalgebra of superderivations of a Grassmann algebra, say , which stands for V (the “standard representation”). It so happens that such a -module is finite dimensional, faithful and rational, and in addition it contains an admissible lattice, namely : so everything is in place to construct the corresponding Chevalley supergroup . In [5, Section 5] this construction is explicitly carried out for type ; in this case, the associated lattice of weights is , the full lattice of weights of . One can clearly do the same, along the same lines, for types S, and H still using the standard representation .
More generally, the weight lattices Λ between and are in bijections with the sublattices of the quotient : but then – see [5, Section 4.27] – there are very few possibilities for Λ, namely four cases for type , two cases for type , and just one case for types W, S and . In particular, for the last three cases the construction of Chevalley supergroups with V the standard representation exhausts all possibilities.
(b) The arguments used to prove Lemma 1.1 above also apply to give a similar result for the case when is simple of classical type: one only needs minimal adaptations, actually simplifications, because is reductive (there is no “extra nilpotent part” such as , say). As a consequence, one has a proof of the fact that “Chevalley supergroups of classical type” as considered in , ,  and  actually do exist.
(c) It is proved in [5, Proposition 4.26] that, under mild assumptions, every Chevalley supergroup of Cartan type is a closed supersubgroup of . Actually, these conditions are slightly ill settled in the statement of that proposition: indeed, instead of “Assume that as a -submodule of is a direct summand” one should read “Assume that as a -submodule of is a direct summand with a -free complement” or (what amounts to be the same) “Assume that the -module is free”.
In fact, the extra condition of “having a -free complement” was actually used in the proof of the proposition, but it was not mentioned in the statement itself. By the way, when is local this extra condition automatically holds true, by Kaplansky’s theorem.
2 Splittings for supergroups and Hopf superalgebras
In what follows we need the notion of “splitting” for both supergroups and Hopf superalgebras. We take it from , where further details may be found. Hereafter, we will think of as being a totally even superalgebra.
2.1 Strongly split Hopf superalgebras
Let H be any commutative Hopf -superalgebra. Then is in fact a Hopf ideal of H, hence is a classical (i.e. super but with trivial odd component) commutative Hopf algebra. Moreover, the coproduct of H induces a structure of super left -comodule on H (via the projection ), such that H is a counital super left -comodule -algebra.
Let be the counit map, let , , and consider . Then has a natural structure of a commutative superalgebra, endowed with a natural “augmentation” map (i.e. a -valued morphism of -superalgebras); moreover, the coproduct of induces on a super left -comodule structure, so is a super counital left -comodule -algebra.
The notion of “strongly split” (commutative) Hopf superalgebra, essentially due to Masuoka – as the core idea was already in his papers  and , but the present terminology is borrowed from  – reads as follows: a commutative Hopf superalgebra H as above is said to be strongly split if is -free and there is an isomorphism of super counital left -comodule -algebras. In particular, Masuoka proved that any commutative Hopf superalgebra over is automatically strongly split when is a field whose characteristic is not 2: cf. [8, Theorem 4.5].
2.2 Global splittings for supergroups
Let be an (affine) supergroup over , the Hopf -superalgebra representing it, and , which is a (classical) commutative Hopf -algebra. The affine group-scheme represented by – so that – is called the classical supersubgroup(-scheme) associated with . The projection yields an embedding , so identifies with a closed (super-) subgroup of . Moreover, every closed supersubgroup of which is classical is a closed subgroup of .
Here is now the definition of “globally split supergroups”. Let be an affine -supergroup for which there exists a closed subsupercheme of , stable by the adjoint -action, such that
, hence we look at as a pointed superscheme,
the product in restricts to an isomorphism of pointed left -superschemes (which will be called a (global) splitting of ),
is isomorphic to a totally odd affine superscheme , as a pointed superscheme.
When all this holds, we say that is globally strongly split, or in short that it is gs-split.
As the referee kindly suggested, a very inspiring (and suggestive) alternative terminology might be that of “equivariantly split” supergroup, which stresses the fact the splitting of such a supergroup G is -equivariant; nevertheless, we have adopted here the terminology of  as we quote results from there.
Let be an affine supergroup, defined over a ring , and let be its representing (commutative Hopf) -superalgebra. Then is globally strongly split if and only if the Hopf superalgebra is strongly split. In particular, if is a field whose characteristic is not 2, then is automatically globally strongly split.
Finally, by [5, Corollary 4.22 (c) and Proposition 4.23] one gets the following:
All Chevalley supergroups of Cartan type as in  are gs-split.
3 Algebraic supergroups of Cartan type: Uniqueness
As recalled in Section 1, the main result in  was the construction of the “Chevalley supergroups” of Cartan type: this proved the existence of any possible type of connected algebraic -supergroup whose complexified tangent Lie superalgebra is (finite dimensional) simple of Cartan type. On the other hand, the uniqueness question was addressed in of [5, Section 4.8], devoted to proving that any algebraic supergroup with the above mentioned properties (in particular for its tangent Lie superalgebra) is necessarily isomorphic to some Chevalley supergroup of Cartan type. However, the result and proof presented there were wrong: hereafter I provide a correct (modified) statement and proof, with changes that affect everything from Section 4.38 through Theorem 4.42 in Section 4.8 of .
3.1 Gs-split, -split supergroups of Cartan type and the Uniqueness Theorem
Let be a connected, gs-split -supergroup; we assume for it that its tangent Lie superalgebra be a -form of a complex Lie superalgebra – i.e., there exists a Lie superalgebra over such that and – and this is simple of Cartan type. Moreover, we assume that the classical subgroup of has a -split maximal torus. In short, we say that is a gs-split, -split supergroup of Cartan type. The group of characters of any -split maximal torus in , call it Λ, contains the root lattice, since acts on by the adjoint action: hence Λ is an intermediate lattice lying between and (notation of Section 1).
Now, for any pair as above there exists a Chevalley -supergroup of Cartan type whose associated pair is exactly – cf. Section 1 above. This yields an “Existence Theorem” for gs-split, -split supergroups of Cartan type.
A related “Uniqueness Theorem” was presented in [5, Section 4.8]. However, it was based on a wrong analysis, so it requires crucial amendments. In fact, I shall present a double version of such a result, one holding true for any ring – but requiring a stronger assumption than that concerning – and one that applies only for fields of zero characteristic – for which the above requirement on is enough.
Here comes the first result:
Let be a connected gs-split, -split supergroup of Cartan type, let be its associated pair and let be the Chevalley supergroup of Cartan type whose group is associated with the same pair . Assume in addition that is isomorphic to . Then is isomorphic to .
To begin with, recall that the classical subgroup of the supergroup splits into a semidirect product , where is a connected -split subgroup with and group of characters Λ, and is a connected, unipotent normal subgroup with . In addition, by construction is obtained via a classical procedure “à la Chevalley” – cf. [5, Proposition 4.9] – based on V thought of as a -module, the Kostant form of , etc. Similarly, too is realized via a construction “à la Chevalley”, which roughly speaking “integrates” the nilpotent Lie algebra (linearized through V). These Chevalley constructions for and are realized simultaneously as parts of the overarching procedure which constructs all of . In addition (as a consequence), the adjoint action of onto is uniquely determined by the adjoint action of onto as well as by Λ.
From the isomorphism we get a corresponding decomposition of as a semidirect product, and also that the adjoint action of onto is uniquely determined by the action of onto and by Λ.
Putting all this together we get, in the language of , that one can express this by saying that the “super Harish-Chandra pairs” of and , namely and , are isomorphic. Now, the main result in  is exactly – cf. Theorem 4.3.14 therein – that the (suitably defined) category of “super Harish-Chandra pairs” is equivalent to the category of (fine) gs-split supergroups. But the latter category contains both our supergroups and (by assumption for the former, and by the remark at the beginning of the proof for the latter): therefore, we can conclude that is isomorphic to . ∎
The second result is a direct consequence:
Let be a connected gs-split, -split supergroup of Cartan type, let be its associated pair and let be the Chevalley supergroup of Cartan type associated with . Moreover, assume that is a field of characteristic zero. Then is isomorphic to .
From the proof of Theorem 3.2 above we know that the subgroup of splits as , where is connected, -split reductive with and group of characters Λ, and is connected, unipotent, normal with . Moreover, the (conjugacy) action of onto is entirely encoded by the adjoint action of onto and by the lattice Λ.
On the other hand, the classical subgroup of has tangent Lie algebra , where is a reductive Lie subalgebra and is a nilpotent ideal of (see the proof of Lemma 1.1). Since is a field of characteristic zero, we have a Chevalley decomposition of into a semidirect product , where is the unipotent radical of ; then this is a connected, unipotent normal subgroup with , while is a connected, -split reductive subgroup with and group of characters the lattice Λ.
By classification theory of -split reductive groups, corresponds to the pair ; but , which also is -split reductive, corresponds to the same pair as well, whence there exists an isomorphism . Similarly, in the classification of connected unipotent (algebraic) group-schemes over fields of characteristic zero both and correspond to the same nilpotent Lie algebra (namely ), hence there exists an isomorphism between them. In addition, the (conjugacy) action of onto is again entirely encoded by the adjoint action of onto and by the lattice Λ.
Comparing now and we conclude that both are semidirect products, with pairwise isomorphic factors, and the action of the reductive factor onto the unipotent (normal) one is ruled in the same way; so these semidirect products are isomorphic, i.e. . Then Theorem 3.2 applies, and we find that is isomorphic to . ∎
The same statements as in Theorems 3.2 and 3.3 above also hold true the case when is simple of classical type and is a “Chevalley supergroup” as in ; indeed, one proves them via the same arguments, and in the second case the proof is even simpler, as is reductive. This yields another, more general proof of the fact that “Chevalley supergroups of classical type” are unique up to isomorphism (cf. ).
The author is greatly indebted to Professor Akira Masuoka, whom he sincerely thanks for his many valuable comments. He also thanks the referee, whose precious remarks helped improving the present work.
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Published Online: 2015-11-17
Published in Print: 2016-09-01