Hopf algebras and quantum groups at roots of unity

1993 ◽  
pp. 102-118
Author(s):  
Jürg Fröhlich ◽  
Thomas Kerler
Keyword(s):  
Quantum Groups ◽  
Hopf Algebras ◽  
Roots Of Unity

THE CENTER OF THE COORDINATE ALGEBRAS OF QUANTUM GROUPS AT pα-th ROOTS OF UNITY

1993 ◽  
Vol 07 (20n21) ◽  
pp. 3547-3550
Author(s):  
BENJAMIN ENRIQUEZ
Keyword(s):  
Quantum Groups ◽  
Roots Of Unity

The coordinate algebras of quantum groups at pα-th roots of unity are finite modules over their centers, at least in a suitable completed sense (cf. [E]). We describe their centers in the completed case, and deduce from this the centers of the non-completed algebras. As in the [dCKP] situation, it is generated by its “Poisson” and “Frobenius” parts.

scholarly journals The nodal cubic and quantum groups at roots of unity

2020 ◽  
Vol 14 (2) ◽  
pp. 667-680
Author(s):  
Ulrich Krähmer ◽  
Manuel Martins
Keyword(s):  
Quantum Groups ◽  
Roots Of Unity

scholarly journals INTRODUCTION TO QUANTIZED LIE GROUPS AND ALGEBRAS

1992 ◽  
Vol 07 (25) ◽  
pp. 6175-6213 ◽  
Author(s):  
T. TJIN
Keyword(s):  
Lie Groups ◽  
Quantum Groups ◽  
Hopf Algebras ◽  
Baxter Equation ◽  
Product Decomposition ◽  
R Matrix ◽  
Conformal Field ◽  
Finite Dimensional ◽  
Lie Groups And Algebras ◽  
Poisson Lie Groups

We give a self-contained introduction to the theory of quantum groups according to Drinfeld, highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras, Poisson structures and deformation quantization are also provided. After defining Poisson Lie groups we study their relation to Lie bialgebras and the classical Yang-Baxter equation. Then we explain in detail the concept of quantization for them. As an example the quantization of sl2 is explicitly carried out. Next we show how quantum groups are related to the Yang-Baxter equation and how they can be used to solve it. Using the quantum double construction we explicitly construct the universal R matrix for the quantum sl2 algebra. In the last section we deduce all finite-dimensional irreducible representations for q a root of unity. We also give their tensor product decomposition (fusion rules), which is relevant to conformal field theory.

scholarly journals QUANTUM GROUPS AT ROOTS OF UNITY AND MODULARITY

2006 ◽  
Vol 15 (10) ◽  
pp. 1245-1277 ◽  
Author(s):  
STEPHEN F. SAWIN
Keyword(s):  
Representation Theory ◽  
Quantum Groups ◽  
Roots Of Unity ◽  
Quantum Field ◽  
Root Of Unity ◽  
Basic Representation ◽  
Topological Quantum ◽  
Quantum Version ◽  
Simply Connected ◽  
Connected Lie Group

We develop the basic representation theory of all quantum groups at all roots of unity (that is, for q any root of unity, where q is defined as in [18]), including Harish–Chandra's theorem, which allows us to show that an appropriate quotient of a subcategory gives a semisimple ribbon category. This work generalizes previous work on the foundations of representation theory of quantum groups at roots of unity which applied only to quantizations of the simplest groups, or to certain fractional levels, or only to the projective form of the group. The second half of this paper applies the representation theory to give a sequence of results crucial to applications in topology. In particular, for each compact, simple, simply-connected Lie group we show that at each integer level the quotient category is in fact modular (thus leading to a Topological Quantum Field Theory), we determine when at fractional levels the corresponding category is modular, and we give a quantum version of the Racah formula for the decomposition of the tensor product.

scholarly journals Multi-brace cotensor Hopf algebras and quantum groups

2016 ◽  
Vol 286 (1-2) ◽  
pp. 657-678
Author(s):  
Xin Fang ◽  
Marc Rosso
Keyword(s):  
Quantum Groups ◽  
Hopf Algebras
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