# Transvector algebras in Clifford analysis

David Eelbode, University of Antwerp, Belgium

The starting point for this talk is the observation that classical harmonic and Clifford analysis are related to the representation theory for the Lie algebra $\mathfrak{sl}(2)$ or its orthosymplectic refinement $\mathfrak{osp}(1,2)$. These objects appear quite naturally as the Howe dual for the spin group, which acts on the space of smooth functions on $\mathbb{R}^m$ with values in either the trivial or spinor representation. In recent years it has become clear that Clifford analysis techniques can also be used to study more complicated conformally invariant differential operators (as classified by Fegan in his seminal paper from 1976), which then act on functions on $\mathbb{R}^m$ with values in more complicated representations for the spin group. Despite the fact that the algebraic framework describing the Howe duality in several variables is well-understood, in terms of the symplectic Lie algebra $\mathfrak{sp}(2k)$ or its refinement $\mathfrak{osp}(1,2k)$, we believe that this is not the only algebra which appears naturally within the setting of higher spin analysis. The aim of the lecture therefore, is to introduce the so-called transvector algebras (which are related to Yangians, as introduced in the framework of the Yang-Baxter equation) and to explain how they show up in various aspects of higher dimensional analysis.

# Operator exponentials for the Clifford-Fourier transform

Presenter: David Eelbode, University of Antwerp, Belgium

Joint work with Eckhard Hitzer

This paper deals with the construction of an exponential representation for the Clifford-Fourier transform defined on multi-vector signals. This is done using the Hamiltonian of the harmonic oscillator, and relies on the classical symmetries underlying harmonic and Clifford analysis on $\mathbb{R}^m$.