Elsevier

Journal of Algebra

Volume 573, 1 May 2021, Pages 476-491
Journal of Algebra

A δ-first Whitehead Lemma

https://doi.org/10.1016/j.jalgebra.2020.12.040Get rights and content

Abstract

We prove that δ-derivations of a simple finite-dimensional Lie algebra over a field of characteristic zero, with values in a finite-dimensional module, are either inner derivations, or, in the case of adjoint module, multiplications by a scalar, or some exceptional cases related to sl(2). This can be viewed as an extension of the classical first Whitehead Lemma.

Introduction

Let L be a Lie algebra, V an L-module, with the module action denoted by •, and δ is an element of the base field. Recall that a δ-derivation of L with values in V is a linear map D:LV such thatD([x,y])=δyD(x)+δxD(y) for any x,yL. The set of all such maps for a fixed δ forms a vector space which will be denoted by Derδ(L,V).

In the case V=L, the adjoint module, we speak about just δ-derivation of L. The latter notion generalizes simultaneously the notions of derivation (ordinary derivations are just 1-derivations) and of centroid (any element of the centroid is, obviously, a 12-derivation).

δ-derivations of Lie and other classes of algebras were a subject of an intensive study (see, for example, [4], [5], [6], [7], [12], [9] and references therein; the latter paper is devoted to a more general notion of so-called quasiderivations which we do not discuss here). As a rule, algebras from “nice” classes (simple, prime, Kac–Moody, Lie algebras of vector fields, etc.) possess a very “few” nontrivial δ-derivations. On the other hand, there are very few results, if at all, about δ-derivations with values in modules.

The aim of this paper is to prove that δ-derivations of simple finite-dimensional Lie algebras of characteristic zero, with values in finite-dimensional modules, are, as a rule, just inner 1-derivations. The exceptional cases are identity maps with values in adjoint modules (which are 12-derivations), or are related to sl(2) which, unlike all other simple Lie algebras, possesses nontrivial δ-derivations. (Note the occurrence of the new exceptional values of δ, in addition to the exceptional values δ=1,12,1 previously known from the literature.) The exact statement runs as follows.

Main theorem

Let g be a semisimple finite-dimensional Lie algebra over an algebraically closed field K of characteristic 0, V a finite-dimensional g-module, and δK. Then Derδ(g,V) is nonzero if and only if one of the following holds:

  • (i)

    δ=1, in which case Der1(g,V)V and consists of inner derivations of the form xxv for some vV.

  • (ii)

    δ=2n for some integer n1, or δ=2n+2 for some integer n3, or δ=12, and V is decomposable into the direct sum of irreducible g-modules in such a way that each direct summand of V is a nontrivial irreducible module over exactly one of the simple direct summands of g, and a trivial module over the rest of them.

In the latter case, decomposing g into the direct sum of simple algebras: g=g1gm, and writingV=V11V1k1V21V2k2Vm1Vmkm, where Vij is an irreducible module over gi, and a trivial module over g, i, we have:Der2n(g,V)gisl(2)VijV(n)Dij, where n1, V(n) is the (n+1)-dimensional irreducible sl(2)-module, and Dij is the (n+3)-dimensional vector space of (2n)-derivations of gisl(2) with values in VijV(n), as described in Lemma 9(ii).Der2n+2(g,V)gisl(2)VijV(n)Eij, where n3, Eij is the (n1)-dimensional vector space of 2n+2-derivations of gisl(2) with values in VijV(n), as described in Lemma 9(iii).Der12(g,V)i=1mVijgiKij, where the inner summation is carried over all occurrences of Vij being isomorphic to the adjoint module gi, and Kij is the one-dimensional vector space spanned by this isomorphism, considered as a map giVij.

Since δ-derivations do not change under field extensions, this theorem essentially describes δ-derivations of a semisimple finite-dimensional Lie algebra with values in a finite-dimensional module, over an arbitrary field of characteristic zero. However, the formulation in the case of an arbitrary field would involve forms of algebras and modules in the sl(2)-related cases, and elements of centroid instead of identity maps in the case δ=12, and would be even more cumbersome, so we confine ourselves with the present formulation.

The classical first Whitehead Lemma states that for g and V as in the statement of the main theorem, the first cohomology vanishes: H1(g,V)=0. As the first cohomology is interpreted as the quotient of derivations of g with values in V modulo inner derivations, and ordinary derivations are just 1-derivations, this theorem can be viewed as an extension of the first Whitehead Lemma. The standard proof of the first Whitehead Lemma involves the Casimir operator (see, for example, [8, Chapter III, §7, Lemma 3]) and will not work in the case δ1. Moreover, taken verbatim, the first Whitehead Lemma is not true for arbitrary δ-derivations, as the exceptional cases related to sl(2), and to the value δ=12 show. Therefore we employ a different approach, which, however, amounts to mere straightforward manipulations with the δ-derivation equation (1), and utilizing standard facts about semisimple Lie algebras and their representations (as exposed, for example, in the classical treatises [2] and [8]).

On the other hand, this theorem is a generalization of the result saying that all nontrivial δ-derivations of simple finite-dimensional Lie algebras of characteristic zero are either ordinary derivations (δ=1), or multiple of the identity map (δ=12), or some special family of (1)-derivations in the case of sl(2) (see [5, Corollary 3], [6, Theorem 2 and Corollary 1], or [9, Corollary 4.6]).

Our initial interest in such sort of results stems from [11], where we computed δ-derivations of certain nonassociative algebras which are of interest in physics (what, in its turn, helped to determine symmetric associative forms on these algebras). These algebras have some classical Lie algebras like sl(n) and so(n) as subalgebras, and considering restriction of δ-derivations to these subalgebras, and employing the theorem above, would allow to streamline some of the proofs in [11].

Section snippets

Auxiliary lemmas

The proof of the main theorem consists of a series of simple lemmas. The ground field K is assumed to be arbitrary, unless stated otherwise.

Lemma 1

Let L be a Lie algebra, and let an L-module V be decomposable into a direct sum of submodules: V=iVi. Then for any δK,Derδ(L,V)iDerδ(L,Vi).

(Here and below the direct sum ⊕ is understood in an appropriate category: either vector spaces, or Lie algebra modules, or Lie algebras, what should be clear from the context.)

Proof

The proof is trivial, and repeats the

The case of sl(2)

In this section we shall prove the main theorem in the case of sl(2). Let the characteristic of the ground field be zero, {e,h,e+} be the standard basis of sl(2) with multiplication table[h,e]=2e,[h,e+]=2e+,[e+,e]=h. The algebra sl(2) is Z-graded. We assign to elements of the standard basis the weights 1, 0, −1, respectively.

Let V(n) denote the irreducible (n+1)-dimensional (i.e., of the highest weight n) sl(2)-module with the standard basis {v0,v1,,vn}. The action is given as follows:ev

The case of gsl(2)

In the previous section we proved the main theorem in the case of sl(2) and an irreducible sl(2)-module. Now we are ready to handle the case of any other simple Lie algebra g and an irreducible g-module.

Lemma 10

Let D be a nonzero δ-derivation, δ1, of a simple finite-dimensional Lie algebra g over an algebraically closed field of characteristic zero, gsl(2), with values in a finite-dimensional irreducible g-module V. Then δ=12, V is isomorphic to the adjoint module, and D is a multiple of the identity

Completion of the proof of the main theorem

Lemma 9, Lemma 10 together establish the main theorem stated in the introduction, in the case of a simple g and an irreducible g-module. Now, basing of this, we complete the proof for the general case of a semisimple g and arbitrary g-module.

If δ=1, the statement reduces to the ordinary derivations, and, as noted above, is equivalent to the first Whitehead Lemma about triviality of the first Chevalley–Eilenberg cohomology H1(g,V). So we may assume δ1. Also, obviously, δ0.

Let g=g1gm be the

Open questions: positive characteristic and infinite-dimensional modules

What happens in positive characteristic? As it is commonly known, the general situation in this case is much more complicated. The first Whitehead Lemma does not hold (in a sense, the opposite is true: as proved in [8, Chapter VI, §3, Theorem 2], any finite-dimensional Lie algebra over a field of positive characteristic has a finite-dimensional module with first nonzero cohomology), so there is no point to conjecture that (most of) δ-derivations should be inner derivations. However, this does

Acknowledgements

GAP [13]1 and Maxima [14] were used to verify some of the computations performed in this paper. Arezoo Zohrabi was supported by grant SGS01/PřF/20-21 of the University of Ostrava.

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