# Algebra

## Quantum groups and representation theory

A model of a physical system (classical or quantum-mechanical) is said to be integrable if it possesses enough symmetries to enable it to be, in some suitable sense, exactly solved. These additional symmetries are typically hidden in some way, and their study has led to the discovery of new algebraic structures, in a broad sense, called quantum groups. Representation theory allows us to better understand quantum groups by realising them concretely using matrices or linear maps and studying transformations of the associated spaces.

Our work in this area concerns:

• Kac-Moody Lie algebras and their deformations
• R-matrices and the Yang-Baxter equation
• W-algebras.
• connections to higher and derived structures

## Semigroup theory

A semigroup is just a set together with an associative binary operation. All groups are all also semigroups. An easy example of a non-group semigroup is the set of positive integers under addition. Rich classes of semigroups include, for example, sets of functions with identical domain and range under composition or square matrices under multiplication.

Some topics that we are particularly interested in:

• transformation semigroups
• uncountable semigroups
• combinatorial semigroup theory
• computational semigroup theory.

## Topological algebra

Staff: Dr Yann Peresse

Topological Algebra studies the interplay of Topology and Algebra. Specifically, it concerns algebraic objects which have a topology that makes their operations continuous. The real numbers form a familiar example, having both an algebraic structure (we can add and multiply them) and a topological structure (they have converging sequences and continuous functions). Crucially, these two structures interact nicely: if x is roughly 3 and y is roughly 5, then x+y is roughly 8 and xy roughly 15.

Some topics that we are particularly interested in:

• topological groups, semigroups and inverse semigroups
• the product topology on transformation semigroups

## General Algebra and Applications

Staff: Dr Catarina Carvalho

An algebraic structure is a set together with operations on that set. General, or universal, algebra studies algebraic structures as a whole, e.g. the variety of groups, or the variety of inverse semigroups, and not specific groups or inverse semigroups.
Some topics that we are particularly interested in:

• applications of  general algebra in computational complexity, namely in  the constraint satisfaction problem
• connections with graph theory, namely polymorphisms of graphs and graph homomorphism
• connections with logic, namely finite model theory.