# Algebra @ Ghent University

## Members of our research group

### Tenured

### Postdocs

- Wouter Castryck (in Leuven, 10% voluntary postdoctoral researcher in Ghent)
- Corentin Le Coz

### PhD students

- Jari Desmet
- Harshit J. Motwani
- Michiel Smet (with cosupervisors Hendrik De Bie and Sigiswald Barbier)
- Inga Valentiner-Branth

## Current research topics

Our main research topics are group theory, non-associative algebraic structures, and related geometric and topological structures.

### Structurable algebras, linear algebraic groups and related structures in geometry and group theory

We investigate the connection between a class of non-associative algebras known as *structurable algebras*
and linear algebraic groups (over arbitrary fields) and related group-theoretical structures such as Moufang sets
(this is a class of doubly transitive permutation groups, introduced by Jacques Tits)
and buildings (also introduced by Jacques Tits) and related point-line geometries.
In particular, we make use of Jordan algebras, quadratic forms, and other algebraic objects related to linear algebraic groups.
We are particularly interested in exceptional linear algebraic groups and in the corresponding cohomological invariants.

- Tom De Medts, Richard M. Weiss, Moufang sets and Jordan division algebras, Math. Ann. 335 (2006), no. 2, 415–433.
- Lien Boelaert, Tom De Medts, Anastasia Stavrova, Moufang sets and structurable division algebras, Mem. Amer. Math. Soc. 259 (2019), no. 1245, 90 pp.
- Tom De Medts, Jeroen Meulewaeter, Inner ideals and structurable algebras: Moufang sets, triangles and hexagons, Israel J. Math. (2021), 36 pp, to appear.

### Finite groups, axial algebras and decomposition algebras

An *axial algebra* is a commutative non-associative algebra generated by certain idempotents such that the multiplication
of eigenspaces with respect to each idempotent satisfies a certain fusion law.
The example after which these algebras have been modeled, is the Griess algebra, the automorphism group of which is the Monster group.
We have been investigating connections between axial algebras and 3-transposition groups and we have developed
a theory of modules for axial algebras.
We have recently introduced the more general notion of *decomposition algebras*, allowing for a wider variety of interesting examples.

We have constructed explicit examples of such algebras for Chevalley groups of simply laced type. For E8, for instance, this provides an explicit construction of a commutative non-associative 3875-dimensional algebra. Notice that this provides (initially unexpectedly) a connection with the our research topic on linear algebraic groups mentioned above.

- Tom De Medts, Michiel Van Couwenberghe, Modules over axial algebras, Algebr. Represent. Theory 23 (2020), no. 1, 209-227.
- Tom De Medts, Simon Peacock, Sergey Shpectorov, Michiel Van Couwenberghe, Decomposition algebras and axial algebras, J. Algebra 556 (2020), 287–314.
- Tom De Medts, Michiel Van Couwenberghe, Non-associative Frobenius algebras for simply laced Chevalley groups, Trans. Amer. Math. Soc. 374 (2021), no. 12, 8715–8774.

### Totally disconnected locally compact groups

We have been studying automorphism groups of locally finite trees, in particular the so-called universal groups (with respect to a prescribed local action). We have extended this to the study of universal groups for locally finite right-angled buildings. More recently, we have extended this to right-angled buildings that are not necessarily locally finite, leading to an interesting interplay between local vs. global topological properties.

- Pierre-Emmanuel Caprace, Tom De Medts, Trees, contraction groups, and Moufang sets, Duke Math. J. 162 (2013), no. 13, 2413–2449.
- Tom De Medts, Ana C. Silva, Koen Struyve, Universal groups for right-angled buildings, Groups Geom. Dyn. 12 (2018), 231–287.
- Jens Bossaert, Tom De Medts, Topological and algebraic properties of universal groups for right-angled buildings, Forum Math. 33 (2021), no. 4, 867–888.