Representation theory of associative algebras is the study how abstract algebraic structures such as rings and algebras can be realised as concrete matrices acting on vector spaces. This means we study ways in which elements of an algebra can be interpreted as linear transformations, making the algebra easier to analyse using tools from linear algebra.
By translating abstract problems into matrix form, representation theory helps us study symmetry, solve equations, and model systems in physics and computer science. Key tools include homological methods, quivers, and Auslander-Reiten theory, which help describe the relationships between modules. The field connects linear algebra, module theory, and category theory, and appears in areas like algebraic geometry and physics.
More specifically, our research interests include representation theory of self-injective algebras and Frobenius algebras, higher-dimensional Auslander-Reiten theory, cluster categories and cluster-tilting theory.
Representation theory of Lie algebras concerns the construction and classification of modules over Lie algebras and related structures. This often involves infinite-dimensional representations, which play a central role in areas such as mathematical physics and algebraic geometry. We study how these modules behave, how they can be built from simpler pieces, and how their structure reflects the properties of the underlying algebras.
In non-associative algebra, we study algebraic structures in which the associative law (ab)c = a(bc) is not satisfied. On direction of research is the application of non-associative algebras to ordinary and partial differential equations, differential geometry, and population genetics. The underlying algebra structures arising in these contexts are generally commutative but non-associative, and they are not necessarily in the well-studied classes of Lie, Jordan, or alternative algebras. For example, in the context of genetic algebras, idempotent elements are the formal representation of the populations in equilibrium, whereas algebraic multiplication represents the second-generation distribution formed by crossover. Geometric, analytical, and combinatorial properties of algebra idempotents, their Peirce decompositions and fusion laws play a central role in our study.
