Compatible actions in semi-abelian categories

Abstract

The concept of a pair of compatible actions was introduced in the case of groups by Brown and Loday [6] and in the case of Lie algebras by Ellis [14]. In this article we extend it to the context of semi-abelian categories (that satisfy the Smith is Huq condition). We give a new construction of the Peiffer product, which specialises to the definitions known for groups and Lie algebras. We use it to prove our main result, on the connection between pairs of compatible actions and pairs of crossed modules over a common base object. We also study the Peiffer product in its own right, in terms of its universal properties, and prove its equivalence with existing definitions in specific cases.

Compatible actions of Lie algebras

“…A proof of this result, based on a proposition in [11], is straightforward but a bit involved, and can be found in the paper in preparation [8]. As for the lower square, we can precompose with the regular epimorphism q♭1 M : this shows that the required commutativity is equivalent to the equation…”

“…commute. This can be proved by using the definition of the coproduct actions and the commutativity of diagrams (5) and (8).…”

“…In the first section we recall basic definitions and results. In the second section we show the link between the notions of compatible actions for groups and for Lie algebras, giving the idea of a possible generalization to a semi-abelian category [13] as it will be considered in the paper [8] in preparation; we show that two crossed modules with a common codomain in Lie R induce compatible actions and, in order to prove the converse, we first give an internal construction of the Peiffer product of two Lie algebras and then we endow it with crossed module structures. Lastly, in the third section we show that the coproduct in XMod L (Lie R ) can be obtained through the Peiffer product and we draw some consequences of this result.…”