Averaging Lie bialgebras

Abstract: In this talk, we establish a bialgebra theory for averaging Lie algebras. First we introduce the notion of a quadratic averaging Lie algebra and show that it induces an isomorphism from the adjoint representation to the coadjoint representation. Then we introduce the notion of matched pairs, Manin triples and bialgebras for averaging Lie algebras, and show that Manin triples, bialgebras and certain matched pairs of averaging Lie algebras are equivalent. In particular, we introduce the notion of an averaging operator on a quadratic Rota-Baxter Lie algebra which can induce an averaging Lie bialgebra naturally.Finally, we introduce the notion of the classical Yang-Baxter equation in an averaging Lie algebra whose solutions give rise to averaging Lie bialgebras. We also introduce the notion of relative Rota-Baxter operators on an averaging Lie algebra and averaging pre-Lie algebras, and construct solutions of the classical Yang-Baxter equation in terms of relative Rota-Baxter operators and averaging pre-Lie algebras. This work joint with Yunhe Sheng and Yanqiu Zhou.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar