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 in a graded differential algebra Omega. We define the notion of an operation of a Hopf algebra in a graded differential algebra Omega which is referred to as a -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 of the Hopf algebra is the universal initial object of the category of -operations with connections.