Enveloping Algebras and Cohomology of Leibniz Pairs

TL;DR

This paper introduces the enveloping algebra for Leibniz pairs and demonstrates that their module categories are equivalent, allowing cohomology to be interpreted via Ext-groups over this algebra.

Contribution

It defines the enveloping algebra for Leibniz pairs and links their cohomology to Ext-groups, providing a new algebraic perspective.

Findings

Category of Leibniz pair modules is isomorphic to modules over its enveloping algebra.

Cohomology of Leibniz pairs can be computed using Ext-groups.

Establishes a new algebraic framework for Leibniz pair cohomology.

Abstract

We introduce the enveloping algebra for a Leibniz pair, and show that the category of modules over a Leibniz pair is isomorphic to the category of left modules over its enveloping algebra. Consequently, we show that the cohomology theory for a Leibniz pair introduced by Flato, Gerstenhaber and Voronov can be interpreted by Ext-groups of modules over the enveloping algebra.