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

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

2012-01-10

arxiv.org/abs/1201.2040v2

Mathematics

Quantum Algebra

40 pages. Some misprints corrected and acknowledgements added

Scientific paper

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.

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