Representation theory of Lie algebras, quantum groups and algebraic groups
represent a major area of mathematical research in the twenty-first century with
numerous applications in other areas of mathematics (geometry, number theory,
combinatorics, finite and infinite groups, etc.) and mathematical physics (such as
conformal field theory, statistical mechanics, integrable systems). Current research
topics in representation theory include quantized enveloping and function algebras,
Kac-Moody Lie algebras, Hecke algebras, canonical bases and crystal bases, vertex
operator algebras, Hall algebras, A∞-algebras, quivers, cluster algebras, Hopf al-
gebras, and Artin-Schelter regular algebras. In particular, representation theory of
quantized Kac-Moody Lie algebras and cohomological theories in noncommutative
algebraic geometry have taken the lead not only within research areas in algebra
but also in other areas of mathematics and physics.
There are various approaches to study representation theory. This proceed-
ings is based on two conferences which focused on the algebraic and combinatorial
approaches to the representation theory.
The first conference was held at the National Institute of Advanced Studies,
Bangalore, India during August 12-16, 2010. This was a satellite conference pre-
ceding the International Congress of Mathematicians held at Hyderabad, India.
The second follow-up conference was held at the University of California, Riverside
during May 18-20, 2012. The speakers at each of these conferences were invited to
contribute to this proceedings.
The articles in this proceedings touch upon a broad spectrum of topics including
quantum groups, crystal bases, categorification, toroidal algebras, vertex algebras,
Hecke algebras, Kazhdan-Lusztig bases, etc.
We thank all participants and speakers at both of these conferences for their
participation and valuable contributions. In particular we thank the speakers who
contributed to this proceedings volume. We are grateful to the National Science
Foundation, USA and the International Center for Theoretical Sciences, India for
providing the financial support for these conferences. Finally, we appreciate the help
from the National Institute of Advanced Studies, Bangalore and the University of
California, Riverside during the respective conferences.