The Weil Algebra of a Hopf Algebra I: A Noncommutative Framework

We generalize the notion, introduced by Henri Cartan, of an operation of a Lie algebra g in a graded differential algebra Ω. We define the notion of an operation of a Hopf algebra H in a graded differential algebra Ω which is referred to as a 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 ( H ) of the Hopf algebra H is the universal initial object of the category of H -operations with connections.