A compendium of Lie structures on tensor products

TL;DR

This paper explores various Lie algebra structures on tensor products using a linear-algebraic technique, connecting diverse topics like associative algebras, Hom-Lie structures, and Poisson brackets, and discussing their generalizations and applications.

Contribution

It introduces a unified linear-algebraic approach to compute and analyze Lie structures on tensor products, extending previous methods to a broad range of algebraic contexts.

Findings

Computed low-degree cohomology of current Lie algebras.

Connected Lie structures to associative and Novikov algebras.

Discussed applications to Poisson brackets and Koszul dual operads.

Abstract

We demonstrate how a simple linear-algebraic technique used earlier to compute low-degree cohomology of current Lie algebras, can be utilized to compute other kinds of structures on such Lie algebras, and discuss further generalizations, applications, and related questions. While doing so, we touch upon such seemingly diverse topics as associative algebras of infinite representation type, Hom-Lie structures, Poisson brackets of hydrodynamic type, Novikov algebras, simple Lie algebras in small characteristics, and Koszul dual operads.