Universal Enveloping Algebras of Lie Antialgebras

TL;DR

This paper introduces the concept of enveloping algebras for Lie antialgebras, explores their properties, and establishes a canonical relationship with Lie superalgebras, showing the enveloping algebra is a quotient of the superalgebra's enveloping algebra.

Contribution

It defines enveloping algebras for Lie antialgebras and demonstrates their connection to Lie superalgebras, providing foundational insights in this emerging area.

Findings

Every Lie antialgebra is related to a Lie superalgebra.

The enveloping algebra of a Lie antialgebra is a quotient of the superalgebra's enveloping algebra.

New framework for studying Lie antialgebras in symplectic geometry.

Abstract

Lie antialgebras is a class of supercommutative algebras recently appeared in symplectic geometry. We define the notion of enveloping algebra of a Lie antialgebra and study its properties. We show that every Lie antialgebra is canonically related to a Lie superalgebra and prove that its enveloping algebra is a quotient of the enveloping algebra of the corresponding Lie superalgebra.