scholarly journals DIAGRAMMATIC CONSTRUCTION OF REPRESENTATIONS OF SMALL QUANTUM $$ \mathfrak{sl} $$2

2021 ◽  
Author(s):  
C. BLANCHET ◽  
M. DE RENZI ◽  
J. MURAKAMI
Keyword(s):  
Quantum Group ◽  
Monoidal Category ◽  
Fundamental Representation ◽  
Combinatorial Description ◽  
Root Of Unity ◽  
Small Quantum ◽  
Odd Order

AbstractWe provide a combinatorial description of the monoidal category generated by the fundamental representation of the small quantum group of $$ \mathfrak{sl} $$ sl 2 at a root of unity q of odd order. Our approach is diagrammatic, and it relies on an extension of the Temperley–Lieb category specialized at δ = −q − q−1.

scholarly journals On semi-infinite cohomology of finite-dimensional graded algebras

2010 ◽  
Vol 146 (2) ◽  
pp. 480-496 ◽  
Author(s):  
Roman Bezrukavnikov ◽  
Leonid Positselski
Keyword(s):  
Quantum Group ◽  
General Setting ◽  
Derived Categories ◽  
Graded Algebras ◽  
Root Of Unity ◽  
Finite Dimensional ◽  
Small Quantum ◽  
Definition Of

AbstractWe describe a general setting for the definition of semi-infinite cohomology of finite-dimensional graded algebras, and provide an interpretation of such cohomology in terms of derived categories. We apply this interpretation to compute semi-infinite cohomology of some modules over the small quantum group at a root of unity, generalizing an earlier result of Arkhipov (posed as a conjecture by B. Feigin).

scholarly journals A quasi-Hopf algebra for the triplet vertex operator algebra

2019 ◽  
Vol 22 (03) ◽  
pp. 1950024 ◽  
Author(s):  
Thomas Creutzig ◽  
Azat M. Gainutdinov ◽  
Ingo Runkel
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Group Algebra ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Root Of Unity

We give a new factorizable ribbon quasi-Hopf algebra [Formula: see text], whose underlying algebra is that of the restricted quantum group for [Formula: see text] at a [Formula: see text]th root of unity. The representation category of [Formula: see text] is conjecturally ribbon equivalent to that of the triplet vertex operator algebra (VOA) [Formula: see text]. We obtain [Formula: see text] via a simple current extension from the unrolled restricted quantum group at the same root of unity. The representation category of the unrolled quantum group is conjecturally equivalent to that of the singlet VOA [Formula: see text], and our construction is parallel to extending [Formula: see text] to [Formula: see text]. We illustrate the procedure in the simpler example of passing from the Hopf algebra for the group algebra [Formula: see text] to a quasi-Hopf algebra for [Formula: see text], which corresponds to passing from the Heisenberg VOA to a lattice extension.

scholarly journals Modules over the small quantum group and semi-infinite flag manifold

2005 ◽  
Vol 10 (3-4) ◽  
pp. 279-362 ◽  
Author(s):  
S. Arkhipov ◽  
R. Bezrukavnikov ◽  
A. Braverman ◽  
D. Gaitsgory ◽  
I. Mirkovic
Keyword(s):  
Quantum Group ◽  
Flag Manifold ◽  
Small Quantum

On unrolled Hopf algebras

2018 ◽  
Vol 27 (10) ◽  
pp. 1850053
Author(s):  
Nicolás Andruskiewitsch ◽  
Christoph Schweigert
Keyword(s):  
Quantum Group ◽  
Lie Algebra ◽  
Hopf Algebras ◽  
Drinfeld Module ◽  
Nichols Algebra ◽  
Small Quantum ◽  
Definition Of ◽  
Special Case

We show that the definition of unrolled Hopf algebras can be naturally extended to the Nichols algebra [Formula: see text] of a Yetter–Drinfeld module [Formula: see text] on which a Lie algebra [Formula: see text] acts by biderivations. As a special case, we find unrolled versions of the small quantum group.

scholarly journals A Few Remarks on the Tube Algebra of a Monoidal Category

2018 ◽  
Vol 61 (3) ◽  
pp. 735-758 ◽  
Author(s):  
Sergey Neshveyev ◽  
Makoto Yamashita
Keyword(s):  
Quantum Group ◽  
Monoidal Category ◽  
Compact Quantum Group ◽  
Morita Context ◽  
Tensor Categories ◽  
Drinfeld Double ◽  
Morita Equivalent

AbstractWe prove two results on the tube algebras of rigid C*-tensor categories. The first is that the tube algebra of the representation category of a compact quantum groupGis a full corner of the Drinfeld double ofG. As an application, we obtain some information on the structure of the tube algebras of the Temperley–Lieb categories 𝒯ℒ(d) ford> 2. The second result is that the tube algebras of weakly Morita equivalent C*-tensor categories are strongly Morita equivalent. The corresponding linking algebra is described as the tube algebra of the 2-category defining the Morita context.

scholarly journals Generalized negligible morphisms and their tensor ideals

2022 ◽  
Vol 28 (2) ◽  
Author(s):  
Thorsten Heidersdorf ◽  
Hans Wenzl
Keyword(s):  
Maximal Ideal ◽  
Monoidal Category ◽  
Single Element ◽  
Trace Function ◽  
Geometric Description ◽  
Prime Characteristic ◽  
Tilting Modules ◽  
Root Of Unity ◽  
Simply Connected ◽  
Deligne Categories

AbstractWe introduce a generalization of the notion of a negligible morphism and study the associated tensor ideals and thick ideals. These ideals are defined by considering deformations of a given monoidal category $${\mathcal {C}}$$ C over a local ring R. If the maximal ideal of R is generated by a single element, we show that any thick ideal of $${\mathcal {C}}$$ C admits an explicitly given modified trace function. As examples we consider various Deligne categories and the categories of tilting modules for a quantum group at a root of unity and for a semisimple, simply connected algebraic group in prime characteristic. We prove an elementary geometric description of the thick ideals in quantum type A and propose a similar one in the modular case.

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