Research

Symmetry is a fundamental concept in mathematics, which is described axiomatically by groups and Lie algebras, however in many of their applications, what is actually important is their representation theory. In the study of their representation theory, it is both natural and useful to consider tensor products of representations. Given two representations, their tensor product decomposes as a direct sum of representations, which can be used to generate new representations. This tensor product then endows the representation category of the group or Lie algebra with the structure of a tensor category. 

In the 1980’s, researchers studying subfactors and quantum integrable systems realized that tensor categories not coming from groups or Lie algebras existed. Subfactors are inclusions of a type of operator algebra called von Neumann algebras. The simplest constructions are given by actions of finite groups, however Jones showed that subfactors not related to groups were possible, and that the bimodule structure of the subfactors produced new tensor categories. 

The study of quantum integrable systems led Drinfeld and Jimbo to introduce new types of Hopf algebras called quantum groups. These can be thought of as deformations of Lie algebras that depend on a parameter q. When q is not a root of unity, the representation theory of the quantum group can be considered equivalent to that of a Lie algebra. However at roots of unity, the representation theory becomes more complicated. In that case, the category can be simplified to produce a semisimple finite tensor category, known as a fusion category. After the discovery of these exotic tensor categories, they found new applications in conformal field theory and statistical mechanics, and also provided a rich source of new invariants, such as the Jones polynomial, in knot theory and low-dimensional topology 

The Yang-Baxter equation is central to modern physics, allowing exact solutions of statistical mechanics systems and conformal field theories. Mathematically, it is also important as its solutions are representations of the braid group. Its solutions can be used to construct the Faddeev-Reshetikhin-Takhtadzhyan (FRT) algebra, which can be considered a deformation of the algebra of functions on the variety of n × n matrices. Closely related to the Yang-Baxter equation is the reflection equation, which describes scattering on the half-line. It motivates another deformation called the reflection equation algebra (REA). The REA can be given a ∗- structure, which allows it to be viewed as a deformation of the algebra of functions on the set of n × n Hermitian matrices. 

The REA is not a quantum group, however it instead forms a comodule algebra over both Oq(Un) and Uq(un). This in turn allows the representation category of the REA to form a module category over the representation categories of Oq(Un) and Uq(un).

The project is joint work with Kenny De Commer. Our overall aim is to classify bounded irreducible ∗-representations of the reflection equation algebra, and study the structure of their resulting module category.

The Temperley-Lieb algebras, TLn(q), n ∈ N, q ∈ C, are a family of algebras that were first discovered in the study of spin chains in statistical mechanics. They were rediscovered in the standard invariant of subfactors by Jones, who used them to define the Jones polynomial knot invariant. Finally Jimbo and Martin showed that TLn(q) is the centralizer algebra of the quantum group Uq(sl2). Remarkably, Kauffman showed that the Temperley-Lieb algebra is diagrammatic algebra, i.e. its elements can be considered as diagrams, with multiplication given by joining diagrams together. The representation theory of this algebra is fundamental to its use in physics, and is now well understood. The representation theory depends on the parameter q. When q is generic the algebra is semisimple, however when q is a root of unity the algebra becomes non-semisimple. 

Our aim is to study the representation theory of the infinite Temperley-Lieb algebra,  a generalization of T Ln(q) to infinitely many generators, focusing on understanding the theory of its finitely generated representations. 

In [1], we fully classify its finite dimensional representations, then introduce a family of representations we call infinite link state representations and classify when they are irreducible or indecomposable. We also define a construction of projective indecomposable representations for TLn that generalizes to give extensions of TL∞ representations. 

In [4], we review the classification of positive extremal traces on the generic infinite Temperley-Lieb algebra, and then extend the classification to the non-semisimple root of unity case. As a result, we obtain Hilbert space structures on the full infinite Temperley-Lieb algebra at roots of unity. 

Conformal field theory (CFT) has attracted a huge amount of mathematical interest, with close relations to various other areas of mathematics discovered, including subfactors, quantum groups, knot theory and low-dimensional topology, and various moonshines. At the same time, it has had a great deal of physical interest and use, from condensed matter systems, such as the quantum Hall effect, to string theory, for example the AdS-CFT correspondence. 

There have been two main approaches to the mathematical study of CFTs. The first, algebraic CFT, focuses on nets of von Neumann algebras on a circle. A natural outcome of this description is a subfactor, which is an inclusion of von Neumann algebras with trivial centre. Subfactors can be described and classified using their standard invariant, and in nice cases this invariant allows a reconstruction of the subfactor. Several methods of constructing a standard invariant have been proposed, the most popular of which is the planar algebra, a sort of diagrammatic graded algebra, the simplest example of which is the Temperley-Lieb algebra. The Temperley-Lieb algebra first appeared in statistical mechanics models describing the Hamiltonian of a lattice, but was rediscovered in terms of subfactors, and were a key part of the formulation of the Jones polynomial. Lattice models themselves provide a source of CFTs through some limiting process, and it has been conjectured that there might be a similar limiting process that gives a way to recover a subfactor related CFT from a subfactor planar algebra. 

The second approach to CFTs is vertex operator algebras (VOA), which are a type of infinite algebra of symmetries, that first appeared in relation to a proof of monstrous moonshine. They can be considered equivalent to the first approach through their representation categories. 

While the algebraic approach tends to require various properties such as unitarity, positivedefiniteness and semisimplicity, which occur in what is known as rational CFT, the VOA approach allows more generality, in what’s known as logarithmic CFT. Previously, most focus has been towards rational CFT, however recently there has been increasing interest in logarithmic examples, the simplest of which is the W(p) model, which have non-semisimple representation categories. In turn, there has been equivalences conjectured between these categories and the full representation categories of small quantum groups at even roots of unity, which are sometimes referred to as the representation categories of restricted quantum groups. For example, the W(2) model and restricted Ui(sl2) have been shown to have equivalent representation categories. 

The aim was to construct the centralizer algebra of restricted quantum sl2, and describe it in terms of a planar algebra. This could be considered as an attempt to construct a lattice model corresponding to the CFT, and relate it to the subfactor approach.

The Deligne category is an abstract tensor category that interpolates the representation categories of the general linear group of rank n to allow you to consider Rep(GLt), where t is an arbitrary complex number. For n an integer, the Deligne category is not abelian, however an abelian envelope was constructed from direct limits of representation categories of super Lie algebras by Entova-Aizenbud, Hinich, and Serganova. 

The project is joint work with Inna Entova-Aizenbud. Our aim is to construct a deformation of the Deligne category, such that the deformation commutes with the abelian envelope, so that we obtain an abelian interpolation of Uq(gln).