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

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.

arXiv comment28 pages. Significant modifications. To appear in Commun. Math. Phys