# Clifford Research Group

#### On a simplicial generalization of Gegenbauer polynomials Tim Janssens (UAntwerp)

Abstract:
When looking at the Laplace operator $\Delta$m on $ℝm$ one can, for each $k \in ℕ$, define the space
$ℋ$k($ℝm$,$ℂ$) := $𝒫$k($ℝm$,$ℂ$) $\cap$ ker $\Delta$m
where $𝒫$k($ℝm$,$ℂ$) is the space of the k-homogeneous polynomials on $ℝm$. It is a well-known result that this space is an irreducible $𝔰𝔬\left(m\right)$-representation and one could wonder whether or not there exist $𝔰𝔬$(m-1)-invariant polynomials $H$k(x) in $ℋ$k($ℝm$,$ℂ$). The answer is given by the abstract branching rules and can be expressed in terms of the classic Gegenbauer polynomials. There also exists a ladder formalism with raising operator X such that
$Xk\left[1\right] = c$k Hk(x)
where $c$k $\in$ $ℂ$. There are many applications of this simple fact e.g. one can use X to construct the Gelfand-Tsetlin basis for $ℋ$k($ℝm$,$ℂ$).
The aim of this seminar is to illustrate that an analogue approach exists in the field of higher spin Clifford analysis where we will focus on the space of the simplicial harmonics. We will then explore a possible application of the special functions that arise.

#### Clifford-Fourier transform on hyperbolic space Pan Lian (UGent - Harbin)

Abstract:
In previous work we have introduced the radial Laplace domain technique to compute the Clifford-Fourier kernel in Euclidean space. In this talk, we will introduce a new generalization of the Helgason-Fourier transform using the angular Dirac operator on both the hyperboloid and unit ball models. We will show how the results in the Laplace domain of Euclidean case and the Lobachevsky case are related. The explicit integral kernels of dimension 2 and 4 and the formal generating function of the even dimensional kernels are derived.

#### Quaternionic Hermitean Clifford Analysis David Eelbode

Abstract:
In this lecture we will introduce the framework of QHCA, which can be seen as a refinement of classical Clifford analysis. Whereas the latter is a function theory for an elliptic conformally invariant differential operator, the former deals with a system of equations defined by the Dirac operator and a few 'deformations' (defined in terms of complex structures). Adding equations to a system reduces the symmetries of the system, and we will show how to determine these symmetries and their dual (in the sense of the Howe pair): this leads to a few interesting observations concerning the question "What are the right equations to consider?".

#### Pizzetti formulas as a tool in harmonic analysis Michael Wutzig

Abstract:
The classical Pizzetti formula provides a way to compute spherical integrals of smooth functions by acting with powers of the Laplacian. In their paper "Pizzetti formulae for Stiefel manifolds and applications" Coulembier and Kieburg generalize this concept to Stiefel manifolds and provide useful formulas when dealing with smooth functions of several variables. In this talk we will give a brief introduction to the aforementioned paper. We will give an example of how to apply these results when dealing with spherical harmonics and in particular their reproducing kernels.

#### The Hermitian submonogenic system Dixan Pena Pena (Politecnico di Milano)

Abstract:
Hermitian Clifford analysis revolves around the study of Dirac-like systems in several complex variables and the main concept in this function theory is that of the h-monogenic functions. In this talk we shall introduce a system whose solutions, called Hermitian submonogenic, are a generalization of the h-monogenic functions. For this system we will consider the Cauchy-Kowalevski extension showing that its solutions are determined by their Cauchy data. A vekua-type system that describes all axially symmetric solutions will also be presented.

#### Lie algebras and parabolic subalgebras Matthias Roels (UA)

Abstract:
In this lecture, we will give a brief overview on the structure theory of simple Lie algebras and parabolic subalgebras. In the first part, we will give a geometrical motivation for the subject and explain why these algebras are of importance in the study of invariant differential operators. Then, we will introduce some basic notions of Lie algebras and in particular, we will give a proper definition of a simple Lie algebra. Once that is done, we will introduce concepts such as Cartan subalgebras, roots, Dynkin diagrams, which are necessary in the classification of simple Lie algebras. Throughout this part, we will use the orthogonal Lie algebra so(m), realised as the space of bivectors inside a Clifford algebra equipped with the commutator bracket, as an example. Finally, we will talk about parabolic subalgebras and its relation with graded Lie algebras. The classification of which yields also a classification of generalized manifolds.

#### Construction of extremal projection operators Tim Raeymaekers

Abstract:
Extremal projection operators are used to project objects (mostly functions) onto the simultaneous kernels of a set of operators which have the property that they can be seen as a set of positive root vectors of a simple Lie algebra. In this talk, I will explain how such operators can be constructed by giving some examples in harmonic and Clifford analysis.

#### Spherical monogenics in dimension 3 and discrete orthogonal polynomials Hendrik De Bie

Abstract:
In this educational talk, my main goal is to show how a family of discrete orthogonal polynomials from the Askey scheme naturally arises in the study of spherical monogenics in dimension 3. This connection will be revealed by making explicit the action of the so(3) Lie algebra that preserves the monogenics on a basis constructed using a tower of CK extensions. It will then be shown that different towers of CK extensions lead to different orthogonal bases. The expansion coefficients between two such bases are subsequently expressed using discrete orthogonal polynomials.

#### The Weyl group and BGG resolutions Matthias Roels

Abstract:
This lecture is a continuation of the lecture on parabolic subalgebras (11/03). In the first part of this lecture, we will introduce some basics of representation theory of parabolic subalgebra. Then, we will discuss the Weyl group and its action on the set of roots of a complex semisimple Lie algebra. Finally all of this can will be used to explain what the BGG resolution is. This resolution is naturally dual to a resolution of sections of certain vector bundles and this result can be used to classify invariant operators on homogeneous spaces of type G/P.

#### The Dirac-Dunkl equation related to the symmetric group Roy Oste

Abstract:
The Dirac-Dunkl operator associated to the symmetric group Sn is considered. Its symmetries are found and are shown to generate an algebra. In the three-dimensional case, eigenfunctions of the Dirac-Dunkl operator are obtained using a Cauchy-Kovalevskaia extension theorem. These eigenfunctions, which correspond to Dunkl monogenics, are seen to support finite-dimensional irreducible representations of the symmetry algebra.

#### The Tits-Kantor-Koecher (TKK) construction for Jordan superalgebras and Jordan superpairs Sigiswald Barbier

Abstract:
In this talk, I will give an introduction to Jordan superalgebras and superpairs. Then we will define the necessary ingredients to construct the Tits-Kantor-Koecher Lie superalgebra associated to a Jordan superalgebra or pair. The TKK-construction was used by Kac to classify all simple finite-dimensional Jordan superalgebras over an algebraically closed field of characteristic zero and by Krutelevich to do the same for Jordan superpairs.

#### The Racah problem for the higher rank Bannai-Ito algebra Wouter van de Vijver

Abstract:
In a previous article Hendrik De Bie, Vincent X. Genest and Luc Vinet proposed a generalization of the Bannai-Ito algebra by considering the symmetry algebra associated to the ℤ2n Dirac-Dunkl equation. The Racah problem for this algebra was previously unsolved. By considering the n-fold tensor product of osp(1|2) we find an alternative realization of the generalized Bannai-Ito algebra. This viewpoint allows us to solve the Racah problem and to define multivariate Bannai-Ito polynomials. Our method can equally be applied to the case of the Racah algebra leading to the generalized Racah algebra and polynomials. The connection with the Bannai-Ito case is explained.

#### Radial algebra as an abstract framework for orthogonal and Hermitian Clifford analysis Ali Guzman Adan

Abstract:
The so-called radial algebra is an algebra generated by a set of abstract vectors variables, which generalizes both polynomial and Clifford algebras. In this talk, we discuss the study of the algebra of its endomorphisms. We will define, using different but equivalent approaches, the abstract versions of the Dirac operator and the directional derivatives. By the study of these and other fundamental endomorphisms defined on the radial algebra setting; we obtain an algebraic structure that can be considered as the abstract equivalent of the Hermitian Clifford analysis. Finally, we present some equivalent axiomatic definitions for the Hermitian radial algebra.

#### Clifford-Fourier analysis: Hardy spaces, Paley-Wiener spaces and wavelets Jeff Hogan

Abstract:
There have been a number of attempts to develop a useful analogue of the classical Fourier transform suitable for the analysis of multichannel signals on Rn. The Clifford-Fourier transform (CFT) of Brackx, De Schepper and Sommen has a number of attractive properties which make it a good candidate for such a role. In this talk we discuss recent work done with Andrew Morris and David Franklin (University of Newcastle) on the use of the CFT in the construction of continuous and discrete wavelets on the plane, the characterization of translation-invariant submodules of the space of square-integrable quaternion-valued functions on the plane, and the proof that the Bernstein space coincides with the Paley-Wiener space in higher dimensions.