A δ-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 such that for any . The set of all such maps for a fixed δ forms a vector space which will be denoted by .
In the case , 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 -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 -derivations), or are related to 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 previously known from the literature.) The exact statement runs as follows.
Main theorem Let be a semisimple finite-dimensional Lie algebra over an algebraically closed field K of characteristic 0, V a finite-dimensional -module, and . Then is nonzero if and only if one of the following holds: , in which case and consists of inner derivations of the form for some . for some integer , or for some integer , or , and V is decomposable into the direct sum of irreducible -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 , and a trivial module over the rest of them.
In the latter case, decomposing into the direct sum of simple algebras: , and writing where is an irreducible module over , and a trivial module over , , we have: where , is the -dimensional irreducible -module, and is the -dimensional vector space of -derivations of with values in , as described in Lemma 9(ii). where , is the -dimensional vector space of -derivations of with values in , as described in Lemma 9(iii). where the inner summation is carried over all occurrences of being isomorphic to the adjoint module , and is the one-dimensional vector space spanned by this isomorphism, considered as a map .
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 -related cases, and elements of centroid instead of identity maps in the case , and would be even more cumbersome, so we confine ourselves with the present formulation.
The classical first Whitehead Lemma states that for and V as in the statement of the main theorem, the first cohomology vanishes: . As the first cohomology is interpreted as the quotient of derivations of 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 . Moreover, taken verbatim, the first Whitehead Lemma is not true for arbitrary δ-derivations, as the exceptional cases related to , and to the value 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 (), or multiple of the identity map (), or some special family of -derivations in the case of (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 and 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: . Then for any ,
(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
In this section we shall prove the main theorem in the case of . Let the characteristic of the ground field be zero, be the standard basis of with multiplication table The algebra is -graded. We assign to elements of the standard basis the weights 1, 0, −1, respectively.
Let denote the irreducible -dimensional (i.e., of the highest weight n) -module with the standard basis . The action is given as follows:
The case of
In the previous section we proved the main theorem in the case of and an irreducible -module. Now we are ready to handle the case of any other simple Lie algebra and an irreducible -module.
Lemma 10 Let D be a nonzero δ-derivation, , of a simple finite-dimensional Lie algebra over an algebraically closed field of characteristic zero, , with values in a finite-dimensional irreducible -module V. Then , 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 and an irreducible -module. Now, basing of this, we complete the proof for the general case of a semisimple and arbitrary -module.
If , 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 . So we may assume . Also, obviously, .
Let 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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