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Volume 28, Issue 5

Issues

Corrigendum to: Algebraic supergroups of Cartan type

Fabio Gavarini
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  • Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della ricerca scientifica 1, I-00133 Roma, Italy
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Published Online: 2015-11-17 | DOI: https://doi.org/10.1515/forum-2015-0044

This Erratum corrects the original online version which can be found here: https://doi.org/10.1515/forum-2011-0144

Abstract

In this note I fix a mistake in my previous paper [5]: namely, the result concerning the uniqueness (up to isomorphisms) of such supergroups needs a new formulation and proof. By the same occasion, I explain more in detail the existence result which comes out of the construction of Chevalley supergroups.

Keywords: Algebraic supergroups; Cartan-type Lie superalgebras

MSC 2010: 14M30; 14A22; 17B20

1 Algebraic supergroups of Cartan type: Existence

All notation and terminology throughout this note will be as in [5]. In particular, every supergroup 𝐆 we shall consider will be fine, which means that its tangent Lie algebra functor Lie(𝐆) is of the form ALie(𝐆)(A)=(A𝕜𝔤)0¯ (where A ranges among all commutative superalgebras) for some Lie superalgebra 𝔤 over 𝕜, with the additional requirement that 𝔤1¯ (as 𝕜-module) be free of finite rank.

The main result in [5] was the construction of the “Chevalley supergroups” of Cartan type, denoted 𝐆V as their construction depends on some suitable 𝔤-module V: these are connected algebraic 𝕜-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 [5] might be obscure on this point, since the construction of 𝐆V 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 𝔤 be a complex Lie superalgebra which is simple of Cartan type; let then Lr and Lw be its root lattice and (integral) weight lattice, respectively. As explained in [5, Section 4.5], for any Chevalley supergroup 𝐆V associated with 𝔤 and a suitable 𝔤-module V its group of characters is a lattice Λ:=ΛV that lies between Lr and Lw; indeed, it is the lattice of weights spanned by the weights (for the action of a Cartan subalgebra of 𝔤) 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 Lr and Lw, there exists a finite-dimensional, rational, faithful g-module which contains an admissible lattice M and whose set of weights spans Λ.

Proof.

We start recalling that the even part of 𝔤 is of the form 𝔤0¯=𝔤0𝔤0¯ where 𝔤0 is a reductive Lie subalgebra and 𝔤0¯ 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 𝔤0-module W whose weights span Λ; moreover, such a W contains a lattice N that is “admissible”, which means stable for the Kostant form K(𝔤0) of U(𝔤0) – cf. [1, Chapter VIII, Section 12.7] (taking into account that 𝔤0 is always simple but when 𝔤 is of type W(n), for then it has a one-dimensional center, and [1, Chapter VIII, Section 12.7, Theorem 2] applies again). Since 𝔤0¯ is an ideal in 𝔤0¯ and 𝔤0¯/𝔤0¯𝔤0, the same W is also a 𝔤0¯-module (by scalar extension), the lattice N being again “admissible”, i.e. stable for K(𝔤0¯); of course W is still rational and finite dimensional.

Second, consider V:=Ind𝔤0¯𝔤(W)=U(𝔤)U(𝔤0¯)W: this is a 𝔤-module which is still finite dimensional and rational; moreover, it contains the admissible lattice M:=K(𝔤)K(𝔤0¯)N, and by construction the lattice ΛV spanned by the weights of V is exactly Λ (since we initially assumed that ΛLr, i.e. Λ contains all roots of 𝔤). Finally, as 𝔤 is simple and its action on V is non-trivial, V itself is also faithful, as required. ∎

(a) Every Cartan type simple Lie superalgebra 𝔤 can be realized (or defined, if you wish) as a Lie subsuperalgebra of some 𝔤𝔩(V) for a suitable V – more precisely, as a suitable Lie superalgebra of superderivations of a Grassmann algebra, say 𝕜[ξ1,,ξn], which stands for V (the “standard representation”). It so happens that such a 𝔤-module V:=𝕜[ξ1,,ξn] is finite dimensional, faithful and rational, and in addition it contains an admissible lattice, namely M:=[ξ1,,ξn]: so everything is in place to construct the corresponding Chevalley supergroup 𝐆V. In [5, Section 5] this construction is explicitly carried out for type W(n); in this case, the associated lattice of weights is Lw, the full lattice of weights of 𝔤. One can clearly do the same, along the same lines, for types S, S~ and H still using the standard representation V:=𝕜[ξ1,,ξn].

More generally, the weight lattices Λ between Lr and Lw are in bijections with the sublattices of the quotient Lw/Lr: but then – see [5, Section 4.27] – there are very few possibilities for Λ, namely four cases for type H(2r), two cases for type H(2r+1), and just one case for types W, S and S~. In particular, for the last three cases the construction of Chevalley supergroups 𝐆V 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 𝔤 is simple of classical type: one only needs minimal adaptations, actually simplifications, because 𝔤0¯ is reductive (there is no “extra nilpotent part” such as 𝔤0¯, say). As a consequence, one has a proof of the fact that “Chevalley supergroups of classical type” as considered in [2], [3], [4] and [6] actually do exist.

(c) It is proved in [5, Proposition 4.26] that, under mild assumptions, every Chevalley supergroup of Cartan type 𝐆V is a closed supersubgroup of GL(V). Actually, these conditions are slightly ill settled in the statement of that proposition: indeed, instead of “Assume that 𝔤1¯ as a 𝕜-submodule of 𝔤𝔩(V)1¯ is a direct summand” one should read “Assume that 𝔤1¯ as a 𝕜-submodule of 𝔤𝔩(V)1¯ is a direct summand with a 𝕜-free complement” or (what amounts to be the same) “Assume that the 𝕜-module 𝔤𝔩(V)1¯/𝔤1¯ is free”.

In fact, the extra condition of “having a 𝕜-free complement” was actually used in the proof of the proposition, but it was not mentioned in the statement itself. By the way, when 𝕜 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 [7], where further details may be found. Hereafter, we will think of 𝕜 as being a totally even superalgebra.

2.1 Strongly split Hopf superalgebras

Let H be any commutative Hopf 𝕜-superalgebra. Then JH:=H1¯2H1¯ is in fact a Hopf ideal of H, hence H¯:=H/JH 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 H¯-comodule on H (via the projection HH¯), such that H is a counital super left H¯-comodule 𝕜-algebra.

Let ϵ:H𝕜 be the counit map, let H+:=Ker(ϵ), H0¯+:=H0¯H+, WH:=H1¯/H0¯+H1¯ and consider WH. Then H¯WH has a natural structure of a commutative superalgebra, endowed with a natural “augmentation” map (i.e. a 𝕜-valued morphism of 𝕜-superalgebras); moreover, the coproduct of H¯ induces on H¯WH a super left H¯-comodule structure, so H¯WH is a super counital left H¯-comodule 𝕜-algebra.

The notion of “strongly split” (commutative) Hopf superalgebra, essentially due to Masuoka – as the core idea was already in his papers [8] and [9], but the present terminology is borrowed from [7] – reads as follows: a commutative Hopf superalgebra H as above is said to be strongly split if WH is 𝕜-free and there is an isomorphism ζ:HH¯𝕜WH of super counital left H¯-comodule 𝕜-algebras. In particular, Masuoka proved that any commutative Hopf superalgebra over 𝕜 is automatically strongly split when 𝕜 is a field whose characteristic is not 2: cf. [8, Theorem 4.5].

2.2 Global splittings for supergroups

Let 𝐆 be an (affine) supergroup over 𝕜, H:=𝒪(𝐆) the Hopf 𝕜-superalgebra representing it, and H¯:=H/JH=H0¯/H1¯2, which is a (classical) commutative Hopf 𝕜-algebra. The affine group-scheme 𝐆ev represented by H¯=𝒪(𝐆)¯ – so that 𝒪(𝐆ev)=𝒪(𝐆)¯ – is called the classical supersubgroup(-scheme) associated with 𝐆 . The projection π:H:=𝒪(𝐆)𝒪(𝐆ev)=H¯ yields an embedding j:𝐆ev𝐆, so 𝐆ev identifies with a closed (super-) subgroup of 𝐆. Moreover, every closed supersubgroup of 𝐆 which is classical is a closed subgroup of 𝐆ev.

Here is now the definition of “globally split supergroups”. Let 𝐆 be an affine 𝕜-supergroup for which there exists a closed subsupercheme 𝐆- of 𝐆, stable by the adjoint 𝐆ev-action, such that

  • (a)

    1𝐆𝐆-, hence we look at 𝐆- as a pointed superscheme,

  • (b)

    the product in 𝐆 restricts to an isomorphism 𝐆ev×𝐆-𝐆 of pointed left 𝐆ev-superschemes (which will be called a (global) splitting of 𝐆),

  • (c)

    𝐆- is isomorphic to a totally odd affine superscheme 𝔸𝕜0|d-, as a pointed superscheme.

When all this holds, we say that 𝐆 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 𝐆ev-equivariant; nevertheless, we have adopted here the terminology of [7] as we quote results from there.

The link with “splittability” of (commutative) Hopf superalgebras is the one we could expect (cf. [8, Theorem 4.5], as well as [7, Theorem 3.2.8 and Corollary 3.2.9]):

Let G be an affine supergroup, defined over a ring 𝕜, and let H:=O(G) be its representing (commutative Hopf) 𝕜-superalgebra. Then G is globally strongly split if and only if the Hopf superalgebra O(G) is strongly split. In particular, if 𝕜 is a field whose characteristic is not 2, then G 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 GV as in [5] are gs-split.

3 Algebraic supergroups of Cartan type: Uniqueness

As recalled in Section 1, the main result in [5] 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 [5].

3.1 Gs-split, 𝕜-split supergroups of Cartan type and the Uniqueness Theorem

Let 𝐆 be a connected, gs-split 𝕜-supergroup; we assume for it that its tangent Lie superalgebra 𝔤:=Lie(𝐆) be a 𝕜-form of a complex Lie superalgebra 𝔤 – i.e., there exists a Lie superalgebra 𝔤 over such that 𝔤=𝕜𝔤 and 𝔤=𝔤 – and this 𝔤 is simple of Cartan type. Moreover, we assume that the classical subgroup 𝐆ev of 𝐆 has a 𝕜-split maximal torus. In short, we say that 𝐆 is a gs-split, 𝕜-split supergroup of Cartan type. The group of characters of any 𝕜-split maximal torus in 𝐆ev, call it Λ, contains the root lattice, since 𝐆ev acts on 𝔤 by the adjoint action: hence Λ is an intermediate lattice lying between Lr and Lw (notation of Section 1).

Now, for any pair (𝔤,Λ) as above there exists a Chevalley 𝕜-supergroup 𝐆V of Cartan type whose associated pair is exactly (𝔤,Λ) – cf. Section 1 above. This yields an “Existence Theorem” for gs-split, 𝕜-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 Lie(𝐆) – and one that applies only for fields of zero characteristic – for which the above requirement on Lie(𝐆) is enough.

Here comes the first result:

Let G be a connected gs-split, 𝕜-split supergroup of Cartan type, let (gC,Λ) be its associated pair and let GV be the Chevalley supergroup of Cartan type whose group is associated with the same pair (gC,Λ). Assume in addition that Gev is isomorphic to (GV)ev. Then G is isomorphic to GV.

Proof.

To begin with, recall that the classical subgroup (𝐆V)ev of the supergroup 𝐆V splits into a semidirect product 𝐆V=𝐆0V𝐆0¯V, where 𝐆0V is a connected 𝕜-split subgroup with Lie(𝐆0V)=𝔤0 and group of characters Λ, and 𝐆0¯V is a connected, unipotent normal subgroup with Lie(𝐆0¯V)=𝔤0¯. In addition, by construction 𝐆0V is obtained via a classical procedure “à la Chevalley” – cf. [5, Proposition 4.9] – based on V thought of as a 𝔤0-module, the Kostant form of U(𝔤0), etc. Similarly, 𝐆0¯V too is realized via a construction “à la Chevalley”, which roughly speaking “integrates” the nilpotent Lie algebra 𝔤0¯ (linearized through V). These Chevalley constructions for 𝐆0V and 𝐆0¯V are realized simultaneously as parts of the overarching procedure which constructs all of 𝐆V. In addition (as a consequence), the adjoint action of (𝐆V)ev onto Lie(𝐆V)=𝔤 is uniquely determined by the adjoint action of Lie((𝐆V)ev)=𝔤0¯ onto Lie(𝐆V)=𝔤 as well as by Λ.

From the isomorphism ϕ:𝐆ev(𝐆V)ev we get a corresponding decomposition of 𝐆ev as a semidirect product, and also that the adjoint action of 𝐆ev onto Lie(𝐆)=𝔤 is uniquely determined by the action of Lie(𝐆ev)=𝔤0¯ onto Lie(𝐆)=𝔤 and by Λ.

Putting all this together we get, in the language of [7], that one can express this by saying that the “super Harish-Chandra pairs” of 𝐆V and 𝐆, namely ((𝐆V)ev,𝔤) and (𝐆ev,𝔤), are isomorphic. Now, the main result in [7] 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 𝐆 and 𝐆V (by assumption for the former, and by the remark at the beginning of the proof for the latter): therefore, we can conclude that 𝐆 is isomorphic to 𝐆V. ∎

The second result is a direct consequence:

Let G be a connected gs-split, 𝕜-split supergroup of Cartan type, let (gC,Λ) be its associated pair and let GV be the Chevalley supergroup of Cartan type associated with (gC,Λ). Moreover, assume that 𝕜 is a field of characteristic zero. Then G is isomorphic to GV.

Proof.

From the proof of Theorem 3.2 above we know that the subgroup (𝐆V)ev of 𝐆V splits as 𝐆V=𝐆0V𝐆0¯V, where 𝐆0V is connected, 𝕜-split reductive with Lie(𝐆0V)=𝔤0 and group of characters Λ, and 𝐆0¯V is connected, unipotent, normal with Lie(𝐆0¯V)=𝔤0¯. Moreover, the (conjugacy) action of 𝐆0V onto 𝐆0¯V is entirely encoded by the adjoint action of 𝔤0 onto 𝔤0¯ and by the lattice Λ.

On the other hand, the classical subgroup 𝐆ev of 𝐆 has tangent Lie algebra Lie(𝐆ev)=𝔤0¯=𝔤0𝔤0¯, where 𝔤0 is a reductive Lie subalgebra and 𝔤0¯ is a nilpotent ideal of 𝔤0¯ (see the proof of Lemma 1.1). Since 𝕜 is a field of characteristic zero, we have a Chevalley decomposition of 𝐆ev into a semidirect product 𝐆ev=𝐆0Ru(𝐆0¯), where Ru(𝐆0¯) is the unipotent radical of 𝐆0¯; then this Ru(𝐆0¯) is a connected, unipotent normal subgroup with Lie(Ru(𝐆0¯))=𝔤0¯, while 𝐆0(𝐆/Ru(𝐆0¯)) is a connected, 𝕜-split reductive subgroup with Lie(𝐆0)=𝔤0 and group of characters the lattice Λ.

By classification theory of 𝕜-split reductive groups, 𝐆0 corresponds to the pair (𝔤0,Λ); but 𝐆0V, which also is 𝕜-split reductive, corresponds to the same pair as well, whence there exists an isomorphism 𝐆0𝐆0V. Similarly, in the classification of connected unipotent (algebraic) group-schemes over fields of characteristic zero both 𝐆0¯V and Ru(𝐆0¯) correspond to the same nilpotent Lie algebra (namely 𝔤0¯), hence there exists an isomorphism between them. In addition, the (conjugacy) action of 𝐆0 onto Ru(𝐆0¯) is again entirely encoded by the adjoint action of 𝔤0 onto 𝔤0¯ and by the lattice Λ.

Comparing now (𝐆V)ev and 𝐆ev 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. 𝐆ev(𝐆V)ev. Then Theorem 3.2 applies, and we find that 𝐆 is isomorphic to 𝐆V. ∎

The same statements as in Theorems 3.2 and 3.3 above also hold true the case when 𝔤 is simple of classical type and 𝐆V is a “Chevalley supergroup” as in [3]; indeed, one proves them via the same arguments, and in the second case the proof is even simpler, as 𝔤0¯ is reductive. This yields another, more general proof of the fact that “Chevalley supergroups of classical type” are unique up to isomorphism (cf. [4]).

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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About the article


Received: 2015-03-10

Revised: 2015-07-23

Published Online: 2015-11-17

Published in Print: 2016-09-01


Citation Information: Forum Mathematicum, Volume 28, Issue 5, Pages 1005–1009, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2015-0044.

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