Mathematical Research Seminar - Archive

Axial algebras are non-associative algebras generated by semisimple idempotents, known as axes, whose eigenvectors obey a given fusion law. These algebras have been shown to occur in many areas of mathematics but are of particular interest thanks to their rich relationship to certain groups. Dihedral axial algebras are those generated by two axes and are fundamental to the study of axial algebras in general. We present a new result that classifies in high generality dihedral axial algebra whose fusion laws have three or fewer eigenvalues. This result represents a significant broadening of our understanding of axial algebras.
 

Given a graph X and a group G we may construct a covering graph Cov(X,Z) by means of a function (called a voltage assignment) Z, that maps arcs of the graph X to elements of the group G. The graph Cov(X,Z) is called the regular cover of X arising from the voltage graph (X,Z) and admits a semiregular (fixed point free) group of automorphisms isomorphic to G. Every graph X with a semiregular group of automorphism G can be regarded as the regular cover of the quotient graph X/G with an appropriate voltage assignment. The theory of voltage graphs and their associated regular covers has become an important tool in the study of symmetries of graphs. We present a generalised theory of voltage graphs where G is allowed to be an arbitrary group (not necesarilly semiregular).

Axial algebras are a recently defined type of nonassociative algebras. They are primarily used to realize groups inside the automorphism group of such an algebra. The prominent example is the Griess algebra that was used in 1982 by R.L. Griess to construct the largest sporadic simple group known as the Monster group. Recently, various other groups were realized in this fashion.

In this project, we have introduced modules over axial algebras. If an axial algebra gives rise to a certain group, then the modules of this algebra naturally correspond to representations of (a central extension of) this group.

We will explain the definitions, give examples and provide some insight into recent developments. We will illustrate how the connection between axial algebras and groups can help us to get a better understanding of both. Joint work with Tom De Medts.

Matroids are combinatorial objects designed to capture essential features of the notion of (linear) independence. They show up naturally in several contexts, including algebraic geometry, topology and optimisation. We will introduce an invariant of matroids called the h-vector that arises when studying the matroid in algebraic and topological contexts. We will explain an old conjecture of Stanley about these h-vectors, survey what is known and explain a new connection with discrete geometry. We will not assume previous knowledge of matroid theory.