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 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 Uprolling unrolled quantum groups

2021 ◽  
pp. 2150023
Author(s):  
Thomas Creutzig ◽  
Matthew Rupert
Keyword(s):  
Quantum Group ◽  
Lie Algebra ◽  
Quantum Groups ◽  
Vertex Operator ◽  
Operator Algebras ◽  
Vertex Algebra ◽  
Vertex Operator Algebras ◽  
Roots Of Unity ◽  
Simple Lie Algebra ◽  
Higher Rank

We construct families of commutative (super) algebra objects in the category of weight modules for the unrolled restricted quantum group [Formula: see text] of a simple Lie algebra [Formula: see text] at roots of unity, and study their categories of local modules. We determine their simple modules and derive conditions for these categories being finite, non-degenerate, and ribbon. Motivated by numerous examples in the [Formula: see text] case, we expect some of these categories to compare nicely to categories of modules for vertex operator algebras. We focus in particular on examples expected to correspond to the higher rank triplet vertex algebra [Formula: see text] of Feigin and Tipunin and the [Formula: see text] algebras.

Cohomology of quantum groups at roots of unity

1993 ◽  
Vol 69 (1) ◽  
pp. 179-198 ◽  
Author(s):  
Victor Ginzburg ◽  
Shrawan Kumar
Keyword(s):  
Quantum Groups ◽  
Roots Of Unity
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