Algebras with representable representations

Abstract

Just like group actions are represented by group automorphisms, Lie algebra actions are represented by derivations: up to isomorphism, a split extension of a Lie algebra $B$ by a Lie algebra $X$ corresponds to a Lie algebra morphism $B\to {\mathit {Der}}(X)$ from $B$ to the Lie algebra ${\mathit {Der}}(X)$ of derivations on $X$ . In this article, we study the question whether the concept of a derivation can be extended to other types of non-associative algebras over a field ${\mathbb {K}}$ , in such a way that these generalized derivations characterize the ${\mathbb {K}}$ -algebra actions. We prove that the answer is no, as soon as the field ${\mathbb {K}}$ is infinite. In fact, we prove a stronger result: already the representability of all abelian actions – which are usually called representations or Beck modules – suffices for this to be true. Thus, we characterize the variety of Lie algebras over an infinite field of characteristic different from $2$ as the only variety of non-associative algebras which is a non-abelian category with representable representations. This emphasizes the unique role played by the Lie algebra of linear endomorphisms $\mathfrak {gl}(V)$ as a representing object for the representations on a vector space $V$ .

Associativity and the cosmash product in operadic varieties of algebras

“…For instance, Remark 7.9 indicates a close relationship between cosmash associativity and algebraic coherence, which may well be valid outside the context of varieties of algebras over a field. A similar question makes sense for action accessibility [4], which is known to be equivalent to algebraic coherence in our present context [9]. The fact that there are no non-abelian varieties over a field which are both cosmash associative and locally algebraically cartesian closed [12]-combine the start of the Introduction, Example 7.5 and Example 8.3may also be an instance of a more general result.…”

“…In previous work, characterisations of the variety Lie K of Lie algebras over an infinite field K (such that charpKq ‰ 2) as a subvariety of Alg K were obtained, in essentially two different ways: in [10,11], it is shown that Lie K is the only non-abelian locally algebraically cartesian closed [12] subvariety of Alg K ; and in [9], the variety Lie K is shown to be the only subvariety of Alg K whose actions are representable [3]. So we have two independent categorical descriptions of the variety Lie K , and thus of the Jacobi identity (however, only for anticommutative algebras).…”

A characterisation of Lie algebras using ideals and subalgebras

“…Like the characterisations of the variety of Lie algebras in [12,14], our work uses computer algebra, though in a significantly different way. Comparing the thus obtained characterisations does, however, lead to an intriguing open problem.…”

On the representability of actions of Leibniz algebras and Poisson algebras

“…However, the notion of action representable category has proven to be quite restrictive. For instance, in [8] the authors proved that, if a variety of non-associative algebras (over a field with ) is action representable, then .…”