# Representation theory

I completed a Ph.D. in pure maths at the University of Sydney and the Universidade de São Paulo and two short post-docs at Paris VII Jussieu and TU München (Munich). My research, which I greatly enjoyed, was in the representation theory of Lie algebras, contractions and some category theory and diagram calculus. Below are the fruits of my labours!

### Imaginary Highest-Weight Representation Theory and Symmetric Functions

Affine sl(2) has two distinct highest-weight theories, classical and imaginary, distinguished by the partition of the root system. Futorny pioneered imaginary highest-weight theory in the general affine case, leaving only one problem unsolved : what is the structure of the imaginary Verma module of level zero? In São Paulo, I attempted to answer this question for affine sl(2). The result is an extensive study of a very particular object! The principal ingredient is a realization of the module in terms of the symmetric functions.

Imaginary Highest-Weight Representation Theory and Symmetric Functions, Comm. Algebra, 37(10):3729–3749, 2009. pdf

### Highest-Weight Theory for Truncated Current Lie Algebras

A truncated current Lie algebra is formed as the tensor product of an underlying Lie algebra with a quotient of a polynomial ring. (Truncated current Lie algebras are also called Takiff algebras, or polynomial Lie algebras.) It turns out that these Lie algebras have an elegant highest-weight theory. In this article the reducibility criteria for the Verma modules of the theory are derived via a study of the Shapovalov determinant. The problem is quite tractable, and is tackled in the general case where the underlying Lie algebra has a triangular decomposition and a non-degenerate pairing of the root spaces; this includes the symmetrizable Kac-Moody Lie algebras, the Virasoro algebra and the Heisenberg algebra. This work was patiently encouraged and guided by Yuly Billig during his visit to Sydney in 2006. These Verma modules provide realizations (via the loop-module construction) of the intriguing exp-polynomial modules studied by Billig, Bermann and Zhao, and permit the derivation of character formulae in the non-degenerate case. This summary of the talk provides an overview of the results and techniques.
Highest-Weight Theory for Truncated Current Lie Algebras, J. Algebra, 2011. pdf
Representations of Truncated Current Lie Algebras, B.H. Neumann Prize Article, Austral. Math. Soc. Gaz., 34(5):279–282, 2007. pdf

### Character Formulae for the Category Õ of Chari

One may construct, for any function on the integers, an irreducible module of level zero for affine sl(2), using the values of the function as structure constants. The modules constructed using exponential-polynomial functions parameterise the irreducible objects of the category Õ of Chari with finite-dimensional homogeneous components. In this work, an expression for the formal character of such an “exponential-polynomial module” is derived using the highest-weight theory of truncations of the loop algebra.
A Character Formula for the Category Õ of Modules for Affine sl(2), Int. Math. Res. Not., 2008. pdf

### Polynomial Lie Algebras and Contractions

This work derives the reducibility criterion for the Verma modules of truncated current Lie algebras using the fascinating analytic method of contractions, instead of using pedestrian algebraic methods.
Polynomial Lie Algebras and Contractions, Proceedings of Structures in Lie Representation Theory, Bremen, 2011. link

### Deligne’s category and representations of general linear supergroups

We classify indecomposable summands of mixed tensor powers of the natural representation for the general linear supergroup up to isomorphism. We also give a formula for the characters of these summands in terms of composite supersymmetric Schur polynomials, and give a method for decomposing their tensor products. Along the way, we describe indecomposable objects in and explain how to decompose their tensor products.
Deligne’s category and representations of general linear supergroups (with Jonny Comes), Representation Theory, 2012. pdf

### Ph.D. Thesis

My Ph.D. thesis, submitted in October 2007, relates three distinct works concerning imaginary highest-weight theory and symmetric functions, representations of polynomial Lie algebras and characters of exponential-polynomial modules.
Vyacheslav Futorny was my supervisor in São Paulo; Alexander Molev (principal) and Gus Lehrer (associate) were my supervisors in Sydney.
Representations of Infinite-Dimensional Lie Algebras, Ph.D. thesis, University of Sydney & Universidade de São Paulo, 2007. pdf