Clifford Research Group

Seminars Academic year 2014-2015

The kernel of the Dunkl Dirac operator as a module for the Bannai-Ito algebra
Hendrik De Bie

In this talk I will discuss how the CK or Cauchy-Kowalewska extension procedure can be developed for the Dunkl Dirac operator related to the reflection group Z2m. This will be used to construct an explicit basis for the kernel in dimension three, expressed in terms of Jacobi polynomials. In turn, by determining the symmetries of the Dunkl Dirac operator in dimension three, we obtain an unexpected connection with the Bannai-Ito algebra and with a scalar operator that also factorizes the Dunkl Laplacian. A detailed comparison will be given between the two approaches.
Joint work with Vincent Genest and Luc Vinet (CRM, Montreal).

Richardson-Gaudin integrability at work for strongly correlated quantum systems
Stijn De Baerdemacker (UGent)

The main difference between our classical and the quantum world is the concept of entanglement, or quantum correlations. This means that the system cannot be regarded as the simple sum of its quasi-independent constituents, but requires a complete formulation of the system's quantum state, which is an exponentially expensive task. Some of the most interesting phenomena in Nature can be found in those systems where the correlations are strong (magnetic spin systems, superconductors, ...), and the particles are strongly entangled. This observation is in sharp contrast with the leading computational tools, which are all based on the (computationally cheap) single-particle paradigm (such as Density Functional Theory), so it is no surprise that these tools often fail for strongly correlated systems.
In this presentation, I will show how one can exploit the concept of (Richardson-Gaudin) integrability to capture strong quantum correlations at (polynomially cheap) computational cost. I will touch upon applications in nuclear, condensed matter physics, and quantum chemistry.

Models in Engineering emerging from Mathematics: a view on their usefulness
Clara Ionescu (UGent)

This presentation gives an overview on the emerging tools from mathematics in system modelling and the challenges they bring within applicability. Time domain and frequency domain are presented and the usefulness of these models are exemplified on real life biomedical applications. The presentation addresses all scientists, engineers, who stay abreast latest trends in modelling.

Higher spin operators in Clifford analysis
Tim Raeymaekers (UGent)

This presentation gives an overview of the work I have been doing during the last years. We explain how the polynomial solution space of higher spin Dirac operators can be fully decomposed in irreducible spin representations

The Joseph ideal for the Lie algebra sl(n) and the Lie superalgebra sl(m,n)
Sigiswald Barbier (UGent)

I will first discuss minimal realisations of the classical semisimple Lie algebras in the Weyl algebra as has been done by Joseph. Then I will give an explicit decomposition of the second tensor product of sl(n) (and sl(m,n)). Using this, one can show that for a one-parameter family of ideals in the enveloping algebra, there exist one special value for which the ideal has infinite codimension. For sl(n) this ideal corresponds to the Joseph ideal defined by the minimal realisation.

Introduction to Jordan algebras
Tom De Medts (UGent)

We will give a gentle introduction to Jordan algebras -a certain class of commutative non-associative algebras- aimed at the non-specialist. We will take time to go through definitions and examples (including Jordan algebras of Clifford type, a.k.a. reduced spin type), and we will discuss the very important Peirce decomposition in Jordan algebras. We will also explain how Jordan algebras give rise to Lie algebras through the so-called Tits-Kantor-Koecher construction, and we will briefly mention structurable algebras, which are a generalization of Jordan algebras.

The harmonic transvector algebra in two vector variables
Matthias Roels (UA/UGent)

The decomposition of polynomials in one vector variable into irreducible modules for the orthogonal group is a crucial result in harmonic analysis which makes use of the Howe duality theorem and leads to the study of spherical harmonics. The aim of this talk is to explain how to describe a decomposition of polynomials in two vector variables and to obtain projection operators on each of the irreducible components. To do so, a particular transvector algebra will be used as a new dual partner for the orthogonal group which leads to a generalisation of the classical Howe duality. In the first part of the talk, we are going to explain the classical Howe duality as well as the definition of a transvector algebra. Then, we will describe the actual decomposition of polynomials in two vector variables into irreducible modules.

Slice Fourier transform: definition, properties and corresponding convolutions
Lander Cnudde (UGent)

In this talk, I will give an overview of the construction of a Fourier transform in a slice Clifford context. Starting from an appropriate extension of the Dirac operator, an orthogonal set of Clifford-Hermite functions is defined and imposed as an eigenvector basis of the integral transform. Inspired by the differential properties of the Clifford-Hermite functions, suitable eigenvalues are found which fix the slice Fourier transform completely. According to these eigenvectors and eigenvalues, a closed form for its kernel function could be obtained using the Mehler formula. Finally, this closed expression allows for the definition of corresponding convolutions and the study of their properties.

Towards a higher rank version of the Bannai-Ito algebra
Hendrik De Bie (UGent)

The Bannai-Ito algebra recently received quite a bit of attention as an algebra related to an important class of orthogonal polynomials. It can equally be realized as the algebra of symmetries of the Dunkl Dirac operator in dimension 3. This provides an interesting way of constructing a higher rank version of the Bannai-Ito algebra by considering the Dunkl Dirac in arbitrary dimension, which is the aim of this talk. We will construct the symmetry algebra by considering the set of all intermediate Gamma operators modified by reflections and describe in detail the action on the representation space. The resulting symmetry algebra can be considered as an n-parameter deformation of the -1 limit of O_q(n).

Laplace transform approach for Clifford-Fourier transform
Pan Lian (UGent)

In this talk, I will introduce a method based on the Laplace transform to compute the closed expressions of the Clifford-Fourier kernels and get the generating function of all even dimensional kernels. This method could be applied to the integral transform whose kernels are defined by series in terms of Bessel function and Gengenbauer polynomials.

Reproducing kernels for spherical monogenics
Michael Wutzig (UGent)

On the space of spherical harmonics there exists a reproducing kernel that is given by a Gegenbauer polynomial. By going over to complex variables, one obtains a reproducing kernel expressed as a Jacobi polynomial. In this talk, the space of hermitian monogenics, which is the space of polynomial bihomogeneous null-solutions of two complex conjugated Dirac operators, is considered. The reproducing kernel for this space is obtained and expressed in terms of Jacobi polynomials.

Doubling Hahn polynomials: classification and applications
Roy Oste (UGent)

We examine how two sets of Hahn polynomials Q_n(x;a,b,N) and Q_n(x;a',b',N') can be combined into a single set of (discrete) orthogonal polynomials P_n(x) (n=0,1,...,N+N'+1). Our analysis uses (new and old) shift operator relations for Hahn polynomials, coming from contiguous relations. This investigation gives rise to new classes of (rather pretty) tridiagonal matrices with a closed form spectrum, and has applications in finite quantum oscillators or in linear spin chains.

Symplectic reflection algebras and Dirac cohomology of eigenspace representations
Jing-Song Huang (HKUST)

We study the eigenfunctions of the differential operators with constant coefficients that are invariant under finite linear groups, especially under finite reflection groups in the framework of representation theory of symplectic reflection algebras. We calculate the Dirac cohomology of these eigenspace representations.

Polynomial realisations of Lie (super)algebras associated to Jordan (super)pairs
Sigiswald Barbier (UGent)

We will give a general method to construct polynomial realisations for Lie superalgebras. This construction will be a generalisation of realisations studied by Conze and Joseph. We will consider in particular the case when our Lie superalgebra is the Tits-Kantor-Koecher Lie superalgebra associated to a Jordan superpair. For a Jordan pair coming from a semi-simple Jordan algebra, we recover the representations used by Jan Möllers to construct minimal representations.

Tensor product decompositions of spin representations in Clifford analysis
Tim Raeymaekers (UGent)

In this talk, we investigate the polynomial kernel of arbitrary higher spin Dirac operators, generalisations of the classical Dirac operator. This kernel is a representation of the spin group. We use representation theoretical results to decompose this kernel into irreducible modules, hereby encountering tensor products of irreducible spin representations.

Structural sets and hyperholomorphy in quaternionic and Clifford analysis
Alí Guzmán Adán (UGent)

The talk will give an overview of the a hyperholomorphic function theory associated to an arbitrary structural set (orthonormal basis of R^{n+1}). We are addressing the study of two different kinds of hyperholomorphy simultaneously, which allows to obtain a generalization of several well-known items from the classical setting. We also briefly discuss the Pi operator, the hyperderivation and a higher order Borel-Pompeiu formula in this framework.

Fischer decomposition in fractional Clifford analysis
Nelson Vieira (University of Aveiro)

What is nowadays called (classic) Clifford analysis consists in the establishment of a function theory for functions belonging to the kernel of the Dirac operator. While such functions can very well describe problems of a particle with internal SU(2)-symmetries, higher order symmetries are beyond this theory. Although many modifications (such as Yang-Mills theory) were suggested over the years they could not address the principal problem, the need of a n-fold factorization of the d'Alembert operator. While Dirac operators with fractional derivatives could achieve this they are more difficult to work with. The main reason is that they do not allow a construction of a Howe dual pair. Hereby, the principal problem is not the invariance under a fractional spin group, but the construction of a Super-Lie-algebra osp(1|2) for general fractional Dirac operators. This requires the application of a new approach and new methods, in particular the establishment of fractional Sommen-Weyl relations. In this talk we will present the building blocks of a function theory for fractional Dirac operators based on Gelfond-Leontiev operators of generalized differentiation. These operators were heavily studied in the 1970th and 1980th by a group of mathematicians from Yeveran (Armenia) and allow particular realizations in form of the classical Caputo and Riemann/Liouville fractional derivatives. This is a joint work with P. Cerejeiras, A. Fonseca, and U. Kähler (CIDMA-University of Aveiro).

Stone - Von Neumann Theorem for Heisenberg Lie supergroup
Jean-Philippe Michel (UCLouvain) and Axel de Goursac (UCLouvain)

The classical Stone - Von Neumann Theorem establishes the classification of the unitary irreducible representations of the Heisenberg Lie group. This is a cornerstone of representation theory and harmonic analysis. In this talk, we will present a similar result for arbitrary real forms of the Heisenberg Lie supergroup. This relies on a new definition of Hilbert superspace and, accordingly, a new definition of unitary representation of Lie supergroups. The previous definitions by Varadarajan et al. (used intensively by Salmasian, Neeb, etc...) are particular of ours.
If the theory of representation of Lie superalgebras is failry well-developed by now, the one of unitary representation of Lie supergroups is at an early stage. Recently, in 2010, Salmasian has obtained a satisfactory analog of Stone - Von Neumann Theorem for a particular real form of the Heisenberg Lie supergroup (when the signature of the metric in the odd coordinates is Euclidean). As for other real forms, one can show that they admit no unitary representations except the trivial one. Our new setting allows to remedy to this problem and get a uniform statement for all real forms.

Department of Mathematical Analysis Department of Mathematical Analysis

Faculty of Engineering and Architecture