# Clifford Research Group

## Seminars Academic year 2006-2007

- Friday 17-11-2006

Hendrik De Bie

*Spherical Harmonics and Integration in Superspace*

- Monday 30-10-2006

David Eelbode

*Root systems and orthogonal Lie algebras*

#### Spherical Harmonics and Integration in
Superspace

Hendrik De Bie (UGent)

*Abstract:*

In this talk the classical theory of spherical harmonics in $R^m$ is extended to superspace using techniques from Clifford analysis. After defining a super-Laplace operator and studying some basic properties of polynomial null-solutions of this operator, a new type of integration on the supersphere is introduced by exploiting the formal equivalence with an old result of Pizzetti. This integral is then used to prove orthogonality of spherical harmonics of different degree, Green-like theorems and also an extension of the important Funk-Hecke theorem to superspace. We hope that this last theorem could be the starting point of a theory of spherical integral transforms in superspace, with obvious mathematical and physical applications.

#### Root systems and orthogonal Lie algebras

David Eelbode (UGent)

*Abstract:*

In this seminar we will introduce the root systems, essential tools for understanding the classification of simple complex Lie algebras. In particular, we consider the example of the orthogonal Lie algebra - realized as the space of bivectors in a complex Clifford algebra, endowed with the commutator. We will explicitely describe the root space decomposition for this algebra, and explain how this generalizes from the adjoint representation to arbitrary finite-dimensional representation spaces. Our final aim is to understand the structure of the spinor spaces.