| Postal address: |
Benjamin Wilson School of Mathematics and Statistics F07 University of Sydney NSW 2006 Australia. |
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| Email: | benw@maths.usyd.edu.au |
| Phone: | +61 2 9351 4076 |
| FAX: | +61 2 9351 4534 |
My PhD thesis, entitled Representations of Infinite-Dimensional Lie Algebras, was submitted in October 2007. It relates three distinct works concerning imaginary highest-weight theory and symmetric functions, representations of polynomial Lie algebras and characters of exponential-polynomial modules.
Future directions and past research are both described in this research statement.
Vyacheslav Futorny was my supervisor in São Paulo; Alexander Molev (principal) and Gus Lehrer (associate) were my supervisors in Sydney.
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I am lecturing at the University of Sydney during the first semester of 2008.
Imaginary Highest-Weight 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.
A preprint entitled Imaginary Highest-Weight Representation Theory and Symmetric Functions is available. It was recently accepted by Communications in Algebra.
I gave a talk on this subject at infinite-dimensional Lie algebras conference held at the Harish-Chandra Research Institute in Allahabad, India (December 2005), and more recently at the algebra seminar at the University of Sydney.
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 preprint (see below) 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 recent visit to Sydney.
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.
I spoke on this subject at the Australian Mathematical Society annual meeting in 2006 and more recently at the Lie and Jordan algebras conference in Maresias, Brazil, in honour of Ivan Shestakov. This summary of the talk provides an overview of the results and techniques.
A preprint entitled Highest-Weight Theory for Truncated Current Lie Algebras is available; it contains all the details.
Characters of Exponential-Polynomial Modules
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 O 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 preprint is available.