Projects
Generalized Askey-Wilson and q-Onsager algebras: a quantum algebraic approach to multivariate orthogonal polynomials Ghent University
The Askey-Wilson algebra was introduced as the algebraic structure behind the Askey-Wilson orthogonal polynomials. It is closely related to the q-Onsager algebra, which originates from statistical mechanics. Both algebras appear in the context of superintegrable quantum systems. Such systems are governed by a Hamiltonian which possesses a sufficient number of symmetry operators. These symmetries are invaluable tools to solve the equations of ...
Quantum symmetric spaces, operator algebras and quantum cluster algebras Vrije Universiteit Brussel
The Structure of Graph Product Operator Algebras KU Leuven
Being part of Voiculescu's groundbreaking non-commutative probability theory, free products of operator algebras are ubiquitous in the theory of operator algebras. They can be viewed as a natural operator-algebraic analogue to free products of groups where both constructions are in a certain sense compatible with each other. In the group setting free products can be generalized in terms of Green's graph products of groups. Inspired by this, ...
Cohomological invariants of structurable algebras Ghent University
A common theme in algebra is to understand algebraic structures over arbitrary fields by first studying them over their algebraic closure and then investigating the possible ways to "descend" to the base field again. A typical example occurs in the theory of quadratic forms over an arbitrary field. In order to decide when two given quadratic forms are non-isometric, a useful tool is to define invariants for quadratic forms; typical (easy) ...
Structurable algebras, representation theory and related point-line geometries Ghent University
The goal of the project is to investigate connections between algebraic structures (linear algebraic groups, Jordan pairs, structurable algebras, Lie algebras) and geometric structures (especially point-line geometries, but also so-called root filtration spaces). We will often rely on the representation theory of the underlying groups.
Axial algebras Ghent University
Non-associative algebras play an important role in many areas of mathematics. The most prominent examples are Lie algebras, introduced in the 1930s to study infinitesimal transformations. Other classes of non-associative algebras also proved vary fruitful in other areas; Jordan algebas, for instance, played a crucial role in Zel'manov's solution to the restricted Burnside problem in group theory. The goal of the proposed project is to explore ...
Non-associative algebras for exceptional groups Ghent University
Linear algebraic groups are matrix groups defined by polynomials. In the past century, a lot of research has been done to develop a classification of these algebraic groups. Among the objects of most interest in this theory are the exceptional groups. Though their classification is complete, a lot of questions remain about these mysterious objects. Recently, a class of algebras that have these exceptional groups as symmetries have been ...
Braided quantum groups, actions and von Neumann algebras Vrije Universiteit Brussel
W*-rigidity for twisted group von Neumann algebras KU Leuven
This is a research proposal in rigidity theory for von Neumann algebras. Building upon earlier rigidity and indecomposability results, I will give the first class of II1 factors that are indecomposable in every possible way: even allowing arbitrary amplifications, they cannot be decomposed as II1 factors coming from groups nor group actions, not even twisted by a 2-cocycle. Next, I aim to prove the first superrigidity theorem for group von ...