Di-semisimple Lie algebras and applications in post-Lie algebra structures

Abstract: We call a Lie algebra $\mathfrak g$ di-semisimple if it can be written as a vector space sum $\mathfrak g = \mathfrak s_1 + \mathfrak s_2$, where $\mathfrak s_1$ and $\mathfrak s_2$ are semisimple subalgebras of $\mathfrak g$ and we say that $\mathfrak g$ is strongly di-semisimple if $\mathfrak g$ can be written as a direct vector space sum of semisimple subalgebras. We will show that complex strongly di-semisimple Lie algebras have to be semisimple themselves.

We will then use this result to show that if a pair of complex Lie algebras $(\mathfrak g, \mathfrak n)$ with $\mathfrak g$ semisimple admits a so called post-Lie algebra structure, then $\mathfrak n$ must be isomorphic to $\mathfrak g$.

Joint work with Dietrich Burde and Mina Monadjem.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar