# Seminars Academic Year 2019-2020

- 24 January 2020

Hadewijch De Clercq (Ghent University)

Defining relations for quantum symmetric pair coideals of Kac-Moody type - 10 February 2020

Nicolas Crampé (CNRS, Institut Denis-Poisson, Tours)

Missing label for su(3), Gaudin model and Bethe equations - 28 February 2020

Sam Claerebout (Ghent University)

Fock model and Segal-Bargmann transform for the orthosymplectic Lie superalgebra - 6 March 2020

Alexis Langlois-Rémillard (Ghent University)

The symmetry algebra of the three-dimensional dihedral Dunkl-Dirac equation

## Defining relations for quantum symmetric pair coideals of Kac-Moody type (Hadewijch De Clerq, Ghent University)

*Abstract:*

Classical symmetric pairs consist of a Kac-Moody algebra g together with its subalgebra of fixed points under an involutive automorphism. Quantum analogs of this construction, known as quantum symmetric pairs, replace the fixed point Lie subalgebras of g by one-sided coideal subalgebras of the quantum group U_q(g). In this talk I will explain this construction and give some motivation and examples arising from quantum integrability. Special attention will be drawn to the presentation of these algebras by generators and relations. I will present a novel set of defining relations of q-Serre or q-Dolan-Grady type for these quantum symmetric pair coideals, based on two different methods. One uses a projection technique established by Letzter and Kolb, the other builds on q-binomial identities.

## Missing label for su(3), Gaudin model and Bethe equations (Nicolas Crampé, CNRS, Institut Denis-Poisson, Tours)

*Abstract:*

I introduce the old problem of the missing label in the direct sum decomposition of two irreducible representations of the Lie algebra su(3). Then, I show how the well-known quantum integrable system called Gaudin model provides this missing label. Finally, I discuss about the resolution via the Bethe ansatz of the Gaudin model and how the associated Bethe equations can be solved to give some explicit examples. This talk is based on an on-going work in collaboration with L. Poulain d'Andecy and L. Vinet.

## Fock model and Segal-Bargmann transform for the orthosymplectic Lie superalgebra (Sam Claerebout, Ghent University)

*Abstract:*

The classical Segal-Bargmann is an integral transform between the Schrödinger space of square-integrable functions and the Fock space of holomorphic functions. The Segal-Bargmann transform was reinterpreted by Hilgert-Kobayashi-Möllers-Ørsted as an intertwining operator between realisations on the Schrödinger and the Fock space of the minimal representation of a Lie group. In this talk I will give a generalisation of this approach to superspaces in order to obtain a Segal-Bargmann transform as an integral transform that intertwines the Schrödinger and Fock model for the orthosymplectic Lie superalgebra osp(m,2|2n).

## The symmetry algebra of the three-dimensional dihedral Dunkl-Dirac equation (Alexis Langlois-Rémillard, Ghent University)

*Abstract:*

The Dunkl-Dirac equation is a deformation of the Dirac equation obtained by changing partial derivatives to Dunkl operators. We consider the cases of three-dimensional Dunkl operators associated to a dihedral root system. The symmetry algebra is considered both abstractly by generators and relations and via the explicit realization by angular momenta operators. The study proceeds by constructing ladder operators on quadratic elements of the algebra of symmetry and by examining the representation theory on the specific example of polynomial monogenics of the Dunkl-Dirac equation obtained in explicit forms by a Cauchy-Kovalevskaya extension theorem.

Work with Hendrik De Bie, Roy Oste and Joris Van der Jeugt