Preface

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.

The Editors

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