Algebra @ Ghent University

Members of our research group

Tenured

Tom De Medts

Postdocs

Wouter Castryck (in Leuven, 10% voluntary postdoctoral researcher in Ghent)
Jari Desmet
Sheevy Leibtag

PhD students

Louis Olyslager
Joachim Slembrouck
Michiel Smet (with cosupervisors Hendrik De Bie and Sigiswald Barbier)
Ben Stoffels

Algebra Seminar @ Ghent University

We organize a regular seminar series covering a range of topics broadly related to the group’s research. Everyone interested is welcome to attend.

Current research topics

Our research focuses on groups and non-associative algebras, with particular emphasis on their connections to Lie theory, geometry and topology.

Structurable algebras, graded Lie algebras and exceptional algebraic groups

We investigate the connections between non-associative algebras, linear algebraic groups and Lie algebras. Jordan algebras, structurable algebras and related algebraic structures provide a common framework for understanding many constructions in these areas. We are particularly interested in exceptional algebraic groups over arbitrary fields and in their associated geometric objects, such as buildings, Moufang sets and Moufang polygons.

Our work includes correspondences between Jordan and structurable division algebras and Moufang sets, as well as constructions of geometries from inner ideals of structurable algebras. These connections allow us to study algebraic groups and their geometries through explicit algebraic operations.

A further direction is the study of graded Lie algebras and root-graded groups. We investigate how their structure can be described using cubic norm structures, cubic norm pairs and quadrangular algebras. We also study operator Kantor pairs over commutative rings and Lie algebras graded by the generalized root system AG₂, extending the interplay between non-associative algebra, Lie theory and geometry.

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.
Sigiswald Barbier, Tom De Medts, Michiel Smet, Operator Kantor pairs, J. Algebra 665 (2025), 298–383.
Tom De Medts, Jeroen Meulewaeter, 5 × 5-graded Lie algebras, cubic norm structures and quadrangular algebras, Forum Math. Sigma 13 (2025), e88, 63 pp.
Tom De Medts, Torben Wiedemann, From cubic norm pairs to G₂- and F₄-graded groups and Lie algebras, preprint (2026).

Axial algebras, Frobenius algebras and vertex algebras

Axial algebras are commutative non-associative algebras generated by idempotents whose eigenspace decompositions satisfy prescribed multiplication rules, known as fusion laws. Their motivating example is the 196884-dimensional Griess algebra, whose automorphism group is the Monster. We study axial algebras and the more general framework of decomposition algebras as a way of relating non-associative algebraic structures to finite groups and algebraic groups.

One focus is the structure of axial algebras of Jordan type and their relationship with Jordan algebras and 3-transposition groups. We investigate finite generation, dimension bounds, subalgebras and automorphism groups. Our results include a dimension bound for primitive 4-generated axial algebras of Jordan type, a characterization of Jordan algebras through solid subalgebras, and descriptions of automorphism groups of Matsuo algebras.

Another focus is the construction of non-associative Frobenius algebras associated with algebraic groups. We have constructed explicit examples for simply laced Chevalley groups and obtained descriptions of Chayet–Garibaldi algebras for several exceptional types. These constructions include a 3876-dimensional algebra associated with E₈.

We also investigate connections with vertex algebras. In particular, we have shown that the Chayet–Garibaldi algebras arise naturally from affine vertex algebras, providing a uniform interpretation of their multiplication. This opens further questions about the relationship between vertex algebras, fusion laws and non-associative algebras.

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).
Jari Desmet, Automorphism groups of Matsuo algebras, J. Pure Appl. Algebra 230 (2026), no. 8, 108309.
Tom De Medts, Louis Olyslager, Constructing Chayet–Garibaldi algebras from affine vertex algebras (including the 3876-dimensional algebra for E₈), Adv. Math. 503 (2026), 111189, 25 pp.

Locally compact groups, buildings and dynamics

We study locally compact groups through their actions on geometric spaces and the interplay between their algebraic and topological properties. Trees and buildings provide a rich source of examples, ranging from algebraic groups over local fields to groups with quite different local and global behaviour.

Our work on groups acting on trees includes connections between contraction groups and Moufang sets. We have also developed universal groups for right-angled buildings, where a prescribed local permutation action determines a global automorphism group. We investigate how properties of the local action influence the algebraic structure and topology of these groups.

These constructions have applications to lattice envelopes of right-angled Artin groups. Here we study which locally compact groups can contain a given discrete group as a lattice, and how the geometry of that discrete group constrains the ambient group.

We also investigate algebraic groups over local fields from the viewpoint of topological group theory and dynamics. Questions about continuous homomorphisms, closed images and compactifications connect this work to unitary representations and mixing properties of group actions.

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.
Pierre-Emmanuel Caprace, Tom De Medts, Lattice envelopes of right-angled Artin groups, Amer. J. Math., to appear.
Uri Bader, Elyasheev Leibtag, Homomorphic images of algebraic groups, J. Algebra 656 (2024), 77–117.
Elyasheev Leibtag, Semi-group compactifications of Algebraic Groups, preprint (2024, revised 2026).