Lie antialgebras: premices

TL;DR

This paper introduces Lie antialgebras, a new class of $ ext{Z}_2$-graded commutative algebras that connect commutative and Lie algebras, along with their representations, structures, and classifications.

Contribution

It develops the foundational theory of Lie antialgebras, including representations, central extensions, and classification of simple finite-dimensional cases.

Findings

Defined Lie antialgebras and their properties

Established analogs of Lie-Poisson structures for these algebras

Classified simple finite-dimensional Lie antialgebras

Abstract

The main purpose of this work is to develop the basic notions of the Lie theory for commutative algebras. We introduce a class of $\mathbbZ_2$-graded commutative but not associative algebras that we call ``Lie antialgebras''. These algebras are closely related to Lie (super)algebras and, in some sense, link together commutative and Lie algebras. The main notions we define in this paper are: representations of Lie antialgebras, an analog of the Lie-Poisson bivector (which is not Poisson) and central extensions. We also classify simple finite-dimensional Lie antialgebras.