1201.2040 (Michel Dubois-Violette et al.)

The Weil Algebra of a Hopf Algebra - I - A noncommutative framework    [PDF]

Michel Dubois-Violette, Giovanni Landi

We generalize the notion, introduced by Henri Cartan, of an operation of a
Lie algebra $\mathfrak g$ in a graded differential algebra $\Omega$. Firstly we
construct a natural extension of the above notion from $\mathfrak g$ to its
universal enveloping algebra $U(\mathfrak g)$ by defining the corresponding
operation of $U(\mathfrak g)$ in $\Omega$. We analyse the properties of this
extension and we define more generally 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.

View original: http://arxiv.org/abs/1201.2040