A new approach to braided monoidal categories

2019 ◽  
Vol 60 (1) ◽  
pp. 013510 ◽  
Author(s):  
Tao Zhang ◽  
Shuanhong Wang ◽  
Dingguo Wang
Keyword(s):  
Monoidal Categories ◽  
New Approach ◽  
Braided Monoidal Categories

scholarly journals Quasitriangular Hopf group algebras and braided monoidal categories

2014 ◽  
Vol 64 (4) ◽  
pp. 893-909
Author(s):  
Shiyin Zhao ◽  
Jing Wang ◽  
Hui-Xiang Chen
Keyword(s):  
Group Algebras ◽  
Monoidal Categories ◽  
Braided Monoidal Categories

scholarly journals Instantons and vortices on noncommutative toric varieties

2014 ◽  
Vol 26 (09) ◽  
pp. 1430008 ◽  
Author(s):  
Lucio S. Cirio ◽  
Giovanni Landi ◽  
Richard J. Szabo
Keyword(s):  
Moduli Spaces ◽  
Toric Varieties ◽  
Partition Functions ◽  
Toric Geometry ◽  
Monoidal Categories ◽  
Klein Quadric ◽  
Noncommutative Algebraic Geometry ◽  
Fundamental Matter ◽  
Braided Monoidal Categories ◽  
Noncommutative Gauge Theories

We elaborate on the quantization of toric varieties by combining techniques from toric geometry, isospectral deformations and noncommutative geometry in braided monoidal categories, and the construction of instantons thereon by combining methods from noncommutative algebraic geometry and a quantized twistor theory. We classify the real structures on a toric noncommutative deformation of the Klein quadric and use this to derive a new noncommutative four-sphere which is the unique deformation compatible with the noncommutative twistor correspondence. We extend the computation of equivariant instanton partition functions to noncommutative gauge theories with both adjoint and fundamental matter fields, finding agreement with the classical results in all instances. We construct moduli spaces of noncommutative vortices from the moduli of invariant instantons, and derive corresponding equivariant partition functions which also agree with those of the classical limit.

scholarly journals THE CENTER OF THE CATEGORY OF BIMODULES AND DESCENT DATA FOR NONCOMMUTATIVE RINGS

2012 ◽  
Vol 11 (06) ◽  
pp. 1250102
Author(s):  
A. L. AGORE ◽  
S. CAENEPEEL ◽  
G. MILITARU
Keyword(s):  
Differential Geometry ◽  
Commutative Ring ◽  
Baxter Equation ◽  
Flat Connection ◽  
Dual Basis ◽  
Monoidal Categories ◽  
Noncommutative Rings ◽  
Descent Data ◽  
Weak Center ◽  
Braided Monoidal Categories

Let A be an algebra over a commutative ring k. We compute the center of the category of A-bimodules. There are six isomorphic descriptions: the center equals the weak center, and can be described as categories of noncommutative descent data, comodules over the Sweedler canonical A-coring, Yetter–Drinfeld type modules or modules with a flat connection from noncommutative differential geometry. All six isomorphic categories are braided monoidal categories: in particular, the category of comodules over the Sweedler canonical A-coring A ⊗ A is braided monoidal. We provide several applications: for instance, if A is finitely generated projective over k then the category of left End k(A)-modules is braided monoidal and we give an explicit description of the braiding in terms of the finite dual basis of A. As another application, new families of solutions for the quantum Yang–Baxter equation are constructed: they are canonical maps Ω associated to any right comodule over the Sweedler canonical coring A ⊗ A and satisfy the condition Ω3 = Ω. Explicit examples are provided.

Coquasitriangular Hopf group coalgebras and braided monoidal categories

2011 ◽  
Vol 6 (5) ◽  
pp. 1009-1020 ◽  
Author(s):  
Meiling Zhu ◽  
Huixiang Chen ◽  
Libin Li
Keyword(s):  
Monoidal Categories ◽  
Braided Monoidal Categories

scholarly journals THE HOPF AUTOMORPHISM GROUP AND THE QUANTUM BRAUER GROUP IN BRAIDED MONOIDAL CATEGORIES

2013 ◽  
Vol 12 (06) ◽  
pp. 1250224
Author(s):  
B. FEMIĆ
Keyword(s):  
Automorphism Group ◽  
Hopf Algebra ◽  
Hopf Algebras ◽  
Brauer Group ◽  
Monoidal Categories ◽  
Brauer Groups ◽  
New Information ◽  
Braided Monoidal Categories ◽  
Usual Quantum ◽  
Module Algebras

With the motivation of giving a more precise estimation of the quantum Brauer group of a Hopf algebra H over a field k we construct an exact sequence containing the quantum Brauer group of a Hopf algebra in a certain braided monoidal category. Let B be a Hopf algebra in [Formula: see text], the category of Yetter–Drinfel'd modules over H. We consider the quantum Brauer group [Formula: see text] of B in [Formula: see text], which is isomorphic to the usual quantum Brauer group BQ(k; B ⋊ H) of the Radford biproduct Hopf algebra B ⋊ H. We show that under certain symmetricity condition on the braiding in [Formula: see text] there is an inner action of the Hopf automorphism group of B on the former. We prove that the subgroup [Formula: see text] — the Brauer group of module algebras over B in [Formula: see text] — is invariant under this action for a family of Radford biproduct Hopf algebras. The analogous invariance we study for BM(k; B ⋊ H). We apply our recent results on the latter group and generate a new subgroup of the quantum Brauer group of B ⋊ H. In particular, we get new information on the quantum Brauer groups of some known Hopf algebras.

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