J Prasad Senesi jsenesi@uottawa.ca

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RESEARCH STATEMENT



RESEARCH
REPRESENTATION THEORY   +   LIE ALGEBRAS

     I am currently a postdoctoral researcher at the University of Ottawa, under the supervision of Professors Erhard Neher and Alistair Savage.  
     My current research research concerns the representation theory of Lie algebras, in particular the finite-dimensional representations of infinite-dimensional Lie algebras such as  Kac-Moody algebras and some of their generalizations.  I began the study of these algebras and their representations under the supervision of Professor Vyjayanthi Chari, while a graduate student at the University of California at Riverside.  We started our investigation with the finite-dimensional representations of the twisted loop algebras.  These infinite-dimensional Lie algebras are one of the main ingredients in the description of the affine Kac-Moody algebras.  We were joined in these efforts by Ghislain Fourier, with whom we classified the (isomorphism classes of) finite-dimensional irreducible representations of a twisted loop algebra. Together we published the results in the Journal of Algebra.  The preprint can be found here:

Weyl modules for the twisted loop algebras
                                                                                                                        (V. Chari, G. Fourier, P. Senesi)

As the title suggests, our paper addresses more than just the irreducible representations.   Extending the earlier work of Chari and Pressley on the representations of the untwisted loop algebras, we defined and classified the Weyl modules for the twisted loops.   These Weyl modules are universal among all loop-highest weight representations.  In particular their classification yields the classification of the irreducible representations.  
     There is more to the category of finite-dimensional representations of a twisted loop algebra than just the irreducible representations, however; this category is non-semisimple, which just means that there are representations that are indecomposable yet reducible.  Extending the work of Prof. Chari and Adriano Moura, I found a decomposition of this category into indecomposable abelian subcategories.  A preprint of these results can be found here:

A block decomposition of finite-dimensional
representations of twisted loop algebras
                                                                                        (P. Senesi)
   
     There are many directions in which one could generalize these results.  For example, there are at least several natural generalizations of the twisted loop algebra:

Twisted Loop Algebra ------> Multiloop Algebra
A twisted loop algebra is a tensor product of a simple Lie algebra with a ring of Laurent polynomials in a single variable.  We may generalize this construction by forming a tensor product of a Lie algebra with a ring of laurent polynomials in several variables.
Twisted Loop Algebra ------> Extended Affine Lie Algebra
Central extensions of the multiloop algebras (above) offer some examples of these generalizations of the affine Kac-Moody algebras.  There are many recent publications devoted to the classification and structure of these algebras.  
Twisted Loop Algebra ------> Quantized Universal Enveloping Algebra
The Weyl modules mentioned above are q --> 1 specializations of  modules for the quantized universal enveloping algebra of the (untwisted) loop algebra.  
Twisted Loop Algebra ------> Lie algebras of regular functions on other varieties, manifolds
The twisted loop algebra may be realized as a Lie algebra of regular functions from the (coordinate ring of) a complex torus to a simple Lie algebra.  The complex torus can be replaced with some other variety, and the resulting structure and representation theory can be studied.


SEMINARS
    I currently attend...
      While a graduate student at UC Riverside,  I enjoyed regularly attending

CONFERENCE PHOTOS
Here's an early photo of me (2nd from right) in 1999 at a regional workshop at the University of Nebraska-Lincoln.    
At the graduate student research conference on algebra and representation theory at Kansas State University in 2006, with (from left to right) Sooraj Kuttykrishnan, Craig Huneke, and Prasad Senesi.  
At the Oberwolfach-Seminar on representation theory and applications, 2006. 
At the Fields Institute 2007 Lie algebra workshop at University of Ottawa.
At the representations of Kac-Moody algebras and combinatorics workshop at BIRS, Banff, Alberta, 2005.
At the quantum affine Lie algebras, extended affine Lie algebras, and applications workshop at BIRS, Banff, Alberta, 2008.



LINKS
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