Algebra @ Ghent University

Members of our research group

Tenured

• Tom De Medts

Postdocs

• Wouter Castryck (in Leuven, 10% voluntary postdoctoral researcher in Ghent)
• Lam Pham
• Arne Van Antwerpen

PhD students

• Jari Desmet
• Louis Olyslager
• Michiel Smet (with cosupervisors Hendrik De Bie and Sigiswald Barbier)
• Inga Valentiner-Branth

Master students

• Lennert De Baecke
• Milan Morreel
• Joachim Slembrouck

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 the corresponding (graded) Lie algebras. In addition, we lay connections with geometric objects known as buildings (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 over arbitrary fields.

• 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. 259 (2024), 33–88.

Axial algebras and decomposition algebras for finite groups and algebraic groups

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 famous 196884-dimensional Griess algebra, the automorphism group of which is the Monster group. We have been investigating connections between axial algebras and 3-transposition groups. More recently, we have introduced the more general notion of decomposition algebras, allowing for a wider variety of interesting examples and fitting into a nicer categorical framework.

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 3876-dimensional algebra. This construction had also been obtained in a different and more general setting by Maurice Chayet and Skip Garibaldi. We have picked up their approach and found an explicit description for many of these Chayet–Garibaldi algebras. Very recently, we have also discovered that these algebras arise naturally inside affine vertex algebras.

Another aspect of our research is the study of primitive axial algebras of Jordan type. We have obtained promising results related to finitely generated algebras in this class. In addition, we have obtained structural results that could potentially lead to a complete classification of such algebras.

• 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.
• Tom De Medts, Louis Rowen, Yoav Segev, Primitive 4-generated axial algebras of Jordan type, Proc. Amer. Math. Soc. 152 (2024), no. 2, 537–551.
• Jari Desmet, Solid lines in axial algebras of Jordan type ½ and Jordan algebras, Proc. Amer. Math. Soc. (2025), available online.
• Tom De Medts, Louis Olyslager, Constructing Chayet–Garibaldi algebras from affine vertex algebras (including the 3876-dimensional algebra for E₈), in preparation.

High-dimensional expanders

The notion of expander graphs can be generalized in a few (non-equivalent) ways to higher dimensions, leading to the notion of a high-dimensional expander (HDX). Our research in this area is conducted through a joint EOS-project with Pierre-Emmanuel Caprace and Timothée Marquis (UCLouvain). We focus on the connection with a class of groups that we called Kac–Moody–Steinberg (KMS) groups.

In our team in Ghent, we have investigated HDX arising from coset complexes over quotients of KMS groups, leading to a systematic generalization of earlier examples of HDX found by Kaufman and Oppenheim.

• Laura Grave de Peralta, Inga Valentiner-Branth, High-dimensional expanders from Kac–Moody–Steinberg groups, Europ. J. Combin., to appear.

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, and later also to right-angled buildings that are not necessarily locally finite, leading to an interesting interplay between local vs. global topological properties. More recently, we have applied specific instances of these universal groups to study lattice envelopes of right-angled Artin groups.

• 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.
• Pierre-Emmanuel Caprace, Tom De Medts, Lattice envelopes of right-angled Artin groups, 33 pp, submitted.