The Weil Algebra of a Hopf Algebra - I - A noncommutative framework

TL;DR

This paper extends the classical concept of Lie algebra operations to Hopf algebras within graded differential algebras, introducing a noncommutative Weil algebra that generalizes algebraic connections.

Contribution

It defines the notion of an $ ext{H}$-operation for Hopf algebras and constructs the universal Weil algebra $W( ext{H})$ as a noncommutative generalization.

Findings

Introduces $ ext{H}$-operations for Hopf algebras in differential algebras

Defines algebraic connections in the noncommutative setting

Constructs the universal Weil algebra $W( ext{H})$

Abstract

We generalize the notion, introduced by Henri Cartan, of an operation of a Lie algebra $\mathfrak g$ in a graded differential algebra $\Omega$. We define the notion of an operation of a Hopf algebra $\mathcal H$ in a graded differential algebra $\Omega$ which is refered to as a $\mathcal H$-operation. We then generalize for such an operation the notion of algebraic connection. Finally we discuss the corresponding noncommutative version of the Weil algebra: The Weil algebra $W(\mathcal H)$ of the Hopf algebra $\mathcal H$ is the universal initial object of the category of $\mathcal H$-operations with connections.