Double Lie algebras of a nonzero weight

TL;DR

This paper introduces $$-double Lie algebras, explores their structure for nonzero weights, confirms a conjecture, and proves the non-existence of simple finite-dimensional cases.

Contribution

It defines $$-double Lie algebras, links them to modified double Poisson algebras, and proves the non-existence of simple finite-dimensional instances.

Findings

$$-double Lie algebras generalize double Lie algebras for nonzero weights

Every $$-double Lie algebra induces a modified double Poisson algebra

No simple finite-dimensional $$-double Lie algebras exist

Abstract

We introduce the notion of $\lambda$-double Lie algebra, which coincides with usual double Lie algebra when $\lambda = 0$. We state that every $\lambda$-double Lie algebra for $\lambda\neq0$ provides the structure of modified double Poisson algebra on the free associative algebra. In particular, it confirms the conjecture of S. Arthamonov (2017). We prove that there are no simple finite-dimensional $\lambda$-double Lie algebras.