- Talks of Representation Theory in Roma "Tor Vergata"
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- Representations of quantum groups at roots of unity, Whittaker vectors and q-W algebras
- Quantum Groups and Their Representations
- Langlands duality for representations of quantum groups

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We first prove several general results about category 0, and then focus on rational Cherednik algebras associated to the general and special linear group over a finite field of the same characteristic as the underlying algebraically closed field. We calculate the characters of irreducible representations with trivial lowest weight of the rational Cherednik algebra associated to GL, Fp, and SL, Fpr , and characters of all irreducible representations of the rational Cherednik algebra associated to GL2 F,.

Description Thesis Ph. Date issued Department Massachusetts Institute of Technology. Publisher Massachusetts Institute of Technology. Collections Mathematics - Ph. Mathematics - Ph. Download all figures.

## Talks of Representation Theory in Roma "Tor Vergata"

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Article activity alert. In mathematics and theoretical physics , the term quantum group denotes various kinds of noncommutative algebras with additional structure.

In general, a quantum group is some kind of Hopf algebra. There is no single, all-encompassing definition, but instead a family of broadly similar objects.

### Representations of quantum groups at roots of unity, Whittaker vectors and q-W algebras

The term "quantum group" first appeared in the theory of quantum integrable systems , which was then formalized by Vladimir Drinfeld and Michio Jimbo as a particular class of Hopf algebra. The same term is also used for other Hopf algebras that deform or are close to classical Lie groups or Lie algebras , such as a "bicrossproduct" class of quantum groups introduced by Shahn Majid a little after the work of Drinfeld and Jimbo.

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Closely related are certain dual objects, also Hopf algebras and also called quantum groups, deforming the algebra of functions on the corresponding semisimple algebraic group or a compact Lie group. Just as groups often appear as symmetries, quantum groups act on many other mathematical objects.

## Quantum Groups and Their Representations

For example, there are quantum planes and quantum Grassmannians. The discovery of quantum groups was quite unexpected since it was known for a long time that compact groups and semisimple Lie algebras are "rigid" objects, in other words, they cannot be "deformed". One of the ideas behind quantum groups is that if we consider a structure that is in a sense equivalent but larger, namely a group algebra or a universal enveloping algebra , then a group or enveloping algebra can be "deformed", although the deformation will no longer remain a group or enveloping algebra.

More precisely, deformation can be accomplished within the category of Hopf algebras that are not required to be either commutative or cocommutative. One can think of the deformed object as an algebra of functions on a "noncommutative space", in the spirit of the noncommutative geometry of Alain Connes.

## Langlands duality for representations of quantum groups

This intuition, however, came after particular classes of quantum groups had already proved their usefulness in the study of the quantum Yang-Baxter equation and quantum inverse scattering method developed by the Leningrad School Ludwig Faddeev , Leon Takhtajan , Evgeny Sklyanin , Nicolai Reshetikhin and Vladimir Korepin and related work by the Japanese School. One type of objects commonly called a "quantum group" appeared in the work of Vladimir Drinfeld and Michio Jimbo as a deformation of the universal enveloping algebra of a semisimple Lie algebra or, more generally, a Kac—Moody algebra , in the category of Hopf algebras.

The resulting algebra has additional structure, making it into a quasitriangular Hopf algebra. There are various coassociative coproducts under which these algebras are Hopf algebras, for example,. Alternatively, the quantum group U q G can be regarded as an algebra over the field C q , the field of all rational functions of an indeterminate q over C. The center of quantum group can be described by quantum determinant. Just as there are many different types of representations for Kac—Moody algebras and their universal enveloping algebras, so there are many different types of representation for quantum groups.

As is the case for all Hopf algebras, U q G has an adjoint representation on itself as a module, with the action being given by.