# Quantum Groups

@inproceedings{Reshetikhin1993QuantumG, title={Quantum Groups}, author={Nicolai Reshetikhin and Theo Johnson-Freyd}, year={1993} }

This thesis consists of four papers. In the first paper we present methods and explicit formulas for describing simple weight modules over twisted generalized Weyl algebras. Under certain conditions we obtain a classification of a class of locally finite simple weight modules from simple modules over tensor products of noncommutative tori. As an application we describe simple weight modules over the quantized Weyl algebra of rank two. In the second paper we derive necessary and sufficient…

## 3,029 Citations

IRREDUCIBLE REPRESENTATIONS OF A CLASS OF FINITE-DIMENSIONAL ALGEBRAS

- Mathematics
- 2002

ABSTRACT In the previous papers1-2 we constructed an -dimensional Hopf algebra which is isomorphic to Drinfeld quantum double of Taft's Hopf algebra if and is a primitive nth root of unity, and…

Gast´on Andrés Garc´ia *

We discuss some general results on finite-dimensional Hopf algebras over an algebraically closed field k of characteristic zero and then apply them to Hopf algebras H of dimension p 3 over k. There…

Modularization of small quantum groups

- Mathematics, Physics
- 2018

We construct a large family of ribbon quasi-Hopf algebras related to small quantum groups, with a factorizable R-matrix. Our main purpose is to obtain non-semisimple modular tensor categories for…

Hopf C*-algebra representations of certain twisted group C*-algebras

- Mathematics
- 1998

In our earlier work, we constructed a specific noncompact quantum group whose quantum group structures have been constructed on a certain twisted group C∗–algebra. Here we study its representation…

On cotriangular Hopf algebras

- Mathematics
- 2000

In an earlier paper, we proved that any triangular semisimple Hopf algebra over an algebraically closed field k of characteristic 0 is obtained from the group algebra k[G] of a finite group G, by…

On modular double of semisimple quantum groups

- Mathematics
- 2018

In this note we propose a construction of the Hopf algebra of a complex analog of devided powers of the Weyl generators of a semisimple simply-laced quantum group. Here we consider the generators as…

Geometric methods in the representation theory of Hecke algebras and quantum groups

- Mathematics, Physics
- 1998

These lectures are mainly based on, and form a condensed survey of the book by N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry, Birkhauser 1997. Various algebras arising…

Induced Representations of Hopf Algebras: Applications to Quantum Groups at Roots of 1

- Mathematics
- 1993

Introduction This paper serves three purposes. The first is to build up a connection between the representation theories of the quantum enveloping algebra U of a semisimple Lie algebra and the…

CYCLIC REPRESENTATIONS OFTHE QUANTUM MATRIX ALGEBRASHANS

- 2007

In this paper we give a complete classiication of the minimal cyclic M q (n)-modules and construct them explicitly. Also, we give a complete classiica-tion of the minimal cyclic modules of the…

Journal of Lie Theory The Quantum Double of a ( locally ) Compact Group

- 1997

We generalise the quantum double construction of Drinfel’d to the case of the (Hopf) algebra of suitable functions on a compact or locally compact group. We will concentrate on the ∗-algebra…

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