A Hopf algebra associated to a Lie pair

TL;DR

This paper constructs a Hopf algebra structure in the derived category for a Lie algebra object derived from a Lie pair, extending the algebraic framework of Lie algebroids.

Contribution

It describes the universal enveloping algebra of a Lie algebra object from a Lie pair and proves it forms a Hopf algebra in the derived category.

Findings

Universal enveloping algebra of the Lie algebra object is a Hopf algebra.

Provides a new algebraic structure associated with Lie pairs.

Extends the theory of Lie algebroids and their algebraic properties.

Abstract

The quotient $L/A[-1]$ of a pair $A\hookrightarrow L$ of Lie algebroids is a Lie algebra object in the derived category $D^b(\mathscr{A})$ of the category $\mathscr{A}$ of left $\mathcal{U}(A)$-modules, the Atiyah class $\alpha_{L/A}$ being its Lie bracket. In this note, we describe the universal enveloping algebra of the Lie algebra object $L/A[-1]$ and we prove that it is a Hopf algebra object in $D^b(\mathscr{A})$.