scholarly journals Classification of Quantum Groups via Galois Cohomology

2019 ◽  
Vol 377 (2) ◽  
pp. 1099-1129
Author(s):  
Eugene Karolinsky ◽  
Arturo Pianzola ◽  
Alexander Stolin
Keyword(s):  
Quantum Groups ◽  
Galois Cohomology

scholarly journals On the classification of easy quantum groups

2013 ◽  
Vol 245 ◽  
pp. 500-533 ◽  
Author(s):  
Moritz Weber
Keyword(s):  
Quantum Groups

scholarly journals ON REPRESENTATIONS OF QUANTUM GROUPS Uq(fm(K,H))

2008 ◽  
Vol 78 (2) ◽  
pp. 261-284 ◽  
Author(s):  
XIN TANG ◽  
YUNGE XU
Keyword(s):  
Quantum Groups ◽  
Irreducible Representations ◽  
Whittaker Model ◽  
Finite Dimensional ◽  
Completely Reducible ◽  
The Relationship ◽  
Natural Families

AbstractWe construct families of irreducible representations for a class of quantum groups Uq(fm(K,H). First, we realize these quantum groups as hyperbolic algebras. Such a realization yields natural families of irreducible weight representations for Uq(fm(K,H)). Second, we study the relationship between Uq(fm(K,H)) and Uq(fm(K)). As a result, any finite-dimensional weight representation of Uq(fm(K,H)) is proved to be completely reducible. Finally, we study the Whittaker model for the center of Uq(fm(K,H)), and a classification of all irreducible Whittaker representations of Uq(fm(K,H)) is obtained.

scholarly journals Classification of globally colorized categories of partitions

2018 ◽  
Vol 21 (04) ◽  
pp. 1850029 ◽  
Author(s):  
Daniel Gromada
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Tensor Category ◽  
Set Partitions

Set partitions closed under certain operations form a tensor category. They give rise to certain subgroups of the free orthogonal quantum group [Formula: see text], the so-called easy quantum groups, introduced by Banica and Speicher in 2009. This correspondence was generalized to two-colored set partitions, which, in addition, assign a black or white color to each point of a set. Globally colorized categories of partitions are those categories that are invariant with respect to arbitrary permutations of colors. This paper presents a classification of globally colorized categories. In addition, we show that the corresponding unitary quantum groups can be constructed from the orthogonal ones using tensor complexification.

scholarly journals QUANTUM GROUPS, q-BOSON ALGEBRAS AND QUANTIZED WEYL ALGEBRAS

2011 ◽  
Vol 22 (05) ◽  
pp. 675-694 ◽  
Author(s):  
XIN FANG
Keyword(s):  
Quantum Groups ◽  
Weyl Algebras ◽  
Conceptual Proof

We give a unified construction of quantum groups, q-boson algebras and quantized Weyl algebras and an action of quantum groups on quantized Weyl algebras. This enables us to give a conceptual proof of the semi-simplicity of the category [Formula: see text] introduced by Nakashima and the classification of all simple objects in it.

scholarly journals Classification of quantum groups and Belavin–Drinfeld cohomologies for orthogonal and symplectic Lie algebras

2016 ◽  
Vol 57 (5) ◽  
pp. 051707 ◽  
Author(s):  
Boris Kadets ◽  
Eugene Karolinsky ◽  
Iulia Pop ◽  
Alexander Stolin
Keyword(s):  
Lie Algebras ◽  
Quantum Groups

On the classification of finite quasi-quantum groups

2017 ◽  
Vol 29 (10) ◽  
pp. 1730003 ◽  
Author(s):  
Mamta Balodi ◽  
Hua-Lin Huang ◽  
Shiv Datt Kumar
Keyword(s):  
Quantum Groups ◽  
Hopf Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Tools And Methods

We give an overview of the classification results obtained so far for finite quasi-quantum groups over an algebraically closed field of characteristic zero. The main classification results on basic quasi-Hopf algebras are obtained by Etingof, Gelaki, Nikshych, and Ostrik, and on dual quasi-Hopf algebras by Huang, Liu and Ye. The objective of this survey is to help in understanding the tools and methods used for the classification.

scholarly journals Construction and classification of \(p\)-ring class fields modulo \(p\)-admissible conductors

2021 ◽  
Vol 5 (1) ◽  
pp. 162-171
Author(s):  
Daniel C. Mayer ◽  
Keyword(s):  
Galois Cohomology ◽  
Base Field ◽  
Class Rank ◽  
Class Field ◽  
Abelian Extensions ◽  
Entire Collection ◽  
Modulo P ◽  
Class Fields ◽  
Insight Into

Each \(p\)-ring class field \(K_f\) modulo a \(p\)-admissible conductor \(f\) over a quadratic base field \(K\) with \(p\)-ring class rank \(\varrho_f\) mod \(f\) is classified according to Galois cohomology and differential principal factorization type of all members of its associated heterogeneous multiplet \(\mathbf{M}(K_f)=\lbrack(N_{c,i})_{1\le i\le m(c)}\rbrack_{c\mid f}\) of dihedral fields \(N_{c,i}\) with various conductors \(c\mid f\) having \(p\)-multiplicities \(m(c)\) over \(K\) such that \(\sum_{c\mid f}\,m(c)=\frac{p^{\varrho_f}-1}{p-1}\). The advanced viewpoint of classifying the entire collection \(\mathbf{M}(K_f)\), instead of its individual members separately, admits considerably deeper insight into the class field theoretic structure of ring class fields. The actual construction of the multiplet \(\mathbf{M}(K_f)\) is enabled by exploiting the routines for abelian extensions in the computational algebra system Magma.

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