Enveloping algebra is a Yetter--Drinfeld module algebra over Hopf algebra of regular functions on the automorphism group of a Lie algebra

TL;DR

This paper constructs a Hopf pairing and module algebra structure linking the universal enveloping algebra of a Lie algebra with functions on its automorphism group, leading to a Hopf algebroid structure.

Contribution

It provides an elementary construction of a Hopf pairing and module algebra structure that results in a Hopf algebroid, extending previous frameworks to include Leibniz algebras.

Findings

Constructed a Hopf pairing between $U(rak{g})$ and $ ext{O}( ext{Aut}(rak{g}))$

Established a braided commutative Yetter--Drinfeld module algebra structure

Developed a Hopf algebroid structure on the smash product algebra

Abstract

We present an elementary construction of a (highly degenerate) Hopf pairing between the universal enveloping algebra $U(\mathfrak{g})$ of a finite-dimensional Lie algebra $\mathfrak{g}$ over arbitrary field $\mathbf{k}$ and the Hopf algebra $\mathcal{O}(\mathrm{Aut}(\mathfrak{g}))$ of regular functions on the automorphism group of $\mathfrak{g}$. This pairing induces a Hopf action of $\mathcal{O}(\mathrm{Aut}(\mathfrak{g}))$ on $U(\mathfrak{g})$ which together with an explicitly given coaction makes $U(\mathfrak{g})$ into a braided commutative Yetter--Drinfeld $\mathcal{O}(\mathrm{Aut}(\mathfrak{g}))$-module algebra. From these data one constructs a Hopf algebroid structure on the smash product algebra $\mathcal{O}(\mathrm{Aut}(\mathfrak{g}))\sharp U(\mathfrak{g})$ retaining essential features from earlier constructions of a Hopf algebroid structure on infinite-dimensional versions of Heisenberg double of $U(\mathfrak{g})$, including a noncommutative phase space of Lie algebra type, while avoiding the need of completed tensor products. We prove a slightly more general result where algebra $\mathcal{O}(\mathrm{Aut}(\mathfrak{g}))$ is replaced by $\mathcal{O}(\mathrm{Aut}(\mathfrak{h}))$ and where $\mathfrak{h}$ is any finite-dimensional Leibniz algebra having $\mathfrak{g}$ as its maximal Lie algebra quotient.