Decompositions of algebras and post-associative algebra structures

Abstract

We introduce post-associative algebra structures and study their relationship to post-Lie algebra structures, Rota-Baxter operators and decompositions of associative algebras and Lie algebras. We show several results on the existence of such structures. In particular we prove that there exists no post-Lie algebra structure on a pair (g, n), where n is a simple Lie algebra and g is a reductive Lie algebra, which is not isomorphic to n. We also show that there is no post-associative algebra structure on a pair (A, B) arising from a Rota-Baxter operator of B, where A is a semisimple associative algebra and B is not semisimple. The proofs use results on Rota-Baxter operators and decompositions of algebras.

Rota–Baxter operators and post-Lie algebra structures on semisimple Lie algebras

“…Hence the proof of Proposition 3.7 and 3.8 is invalid. However, the statement of both results is true and we have given a new proof of it in our paper [16] on decompositions of algebras and post-associative algebra structures. Proposition 3.6.…”

Crystallographic actions on Lie groups and post-Lie algebra structures

“…In case one of the Lie algebras g, n is simple we have the following results. If we interchange the roles of g and n we only can prove the following result, see [35].…”

Semisimple decompositions of Lie algebras and prehomogeneous modules

“…For related results on post-Lie algebra structures see also [4,5]. Furthermore, post-Lie algebra structures are closely related to Rota-Baxter operators, which in turn are naturally related to decomposition results of Lie algebras, see [7,8]. For rigidity results in the context of post-Lie algebra structures on pairs (g, n) of semisimple respectively reductive Lie algebras, decomposition results on the sum of two semisimple Lie algebras are of great importance.…”