Non-abelian tensor product and homology of Lie Superalgebras

TL;DR

This paper introduces the non-abelian tensor product for Lie superalgebras, explores its properties, and connects it to homology theories, providing new tools for understanding their structure and extensions.

Contribution

It defines the non-abelian tensor and exterior products for Lie superalgebras and establishes their homological properties and relationships with cyclic homology.

Findings

Defined the non-abelian tensor product for Lie superalgebras.

Established properties like nilpotency, solvability, and Engel conditions.

Linked homology of Lie superalgebras with cyclic homology of associative superalgebras.

Abstract

We introduce the non-abelian tensor product of Lie superalgebras, study some of its properties including nilpotency, solvability and Engel, and we use it to describe the universal central extensions of Lie superalgebras. We present the low-dimensional non-abelian homology of Lie superalgebras and establish its relationship with the cyclic homology of associative superalgebras. We also define the non-abelian exterior product and give an analogue of Miller's theorem, Hopf formula and a six-term exact sequence for the homology of Lie superalgebras.