Compatible actions of Lie algebras

Abstract

We study compatible actions (introduced by Brown and Loday in their work on the non-abelian tensor product of groups) in the category of Lie algebras over a fixed ring. We describe the Peiffer product via a new diagrammatic approach, which specializes to the known definitions both in the case of groups and in the case of Lie algebras. We then use this approach to transfer a result linking compatible actions and pairs of crossed modules over a common base object L from groups to Lie algebras. Finally, we show that the Peiffer product, naturally endowed with a crossed module structure, has the universal property of the coproduct in XMod L (Lie R ).

Compatible actions in semi-abelian categories

“…Via results in [11], this further implies that under an additional condition called (UA), the actions induced by two L-crossed module structures have a Peiffer product which is again an L-crossed module; furthermore, it is the coproduct in XMod L (A) of the given L-crossed modules. This generalises Proposition 3.4 in [13].…”

“…commute. This can be proved by using diagrams (2), ( 3), ( 11) and (13). In order to prove (8)-we will show only one of the two equalities, since the proof of the other follows the same steps-we are going to use the commutative diagrams induced from the crossed module structures involving L, that is…”

“…We are going to give a definition of compatible internal actions which is inspired by Definitions 3.1 and 3.3, with some differences that we will explain here. • First of all, the commutativity of the two squares in (CA.0) involving the ternary cosmash products is for free in Grp (and in Lie R as one can see from [13]). Right now it is not clear to us what are the conditions the category A must satisfy for the commutativity of these squares to be implied by the other four triangles in (CA.0).…”

“…l M and l N are defined as in(13).Further, as in Proposition 4.3, the compatible actions induced by these crossed module structures coincide with the actions ξ N M and ξ M N . Proof.…”

“…The concept of a pair of compatible actions was first introduced in the category of groups by Brown and Loday, in relation to their work on the non-abelian tensor product of groups [6]. Later, in [14], the definition was adapted to the context of Lie algebras, where it was further studied in [26,13]. Since then, several other particular instances of compatible actions have been defined or used, in various settings: see for example [17,9,8], and [22,23] where the non-abelian derived functors of ⊗ were studied.…”