Quantum group constructions in a symmetric monoidal category

1997 ◽  
Vol 25 (1) ◽  
pp. 117-158 ◽  
Author(s):  
Horia C. Pop
Keyword(s):  
Quantum Group ◽  
Monoidal Category ◽  
Symmetric Monoidal Category

scholarly journals Twin TQFTs and Frobenius Algebras

2013 ◽  
Vol 2013 ◽  
pp. 1-25
Author(s):  
Carmen Caprau
Keyword(s):  
Monoidal Category ◽  
Topological Quantum Field Theory ◽  
Algebra Homomorphism ◽  
Frobenius Algebra ◽  
Quantum Field ◽  
Generators And Relations ◽  
Symmetric Monoidal Category ◽  
Frobenius Algebras ◽  
Topological Quantum ◽  
Algebraic Description

We introduce the category of singular 2-dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on atwin Frobenius algebra, by providing a description of this category in terms of generators and relations. A twin Frobenius algebra(C,W,z,z∗)consists of a commutative Frobenius algebraC, a symmetric Frobenius algebraW, and an algebra homomorphismz:C→Wwith dualz∗:W→C, satisfying some extra conditions. We also introduce a generalized 2-dimensional Topological Quantum Field Theory defined on singular 2-dimensional cobordisms and show that it is equivalent to a twin Frobenius algebra in a symmetric monoidal category.

Monad compositions II: Kleisli strength

2008 ◽  
Vol 18 (3) ◽  
pp. 613-643 ◽  
Author(s):  
ERNIE MANES ◽  
PHILIP MULRY
Keyword(s):  
Monoidal Category ◽  
Linear Equations ◽  
Large Collection ◽  
Data Types ◽  
Cartesian Closed Category ◽  
Closed Category ◽  
Symmetric Monoidal Category ◽  
Cartesian Closed

In this paper we introduce the concept of Kleisli strength for monads in an arbitrary symmetric monoidal category. This generalises the notion of commutative monad and gives us new examples, even in the cartesian-closed category of sets. We exploit the presence of Kleisli strength to derive methods for generating distributive laws. We also introduce linear equations to extend the results to certain quotient monads. Mechanisms are described for finding strengths that produce a large collection of new distributive laws, and consequently monad compositions, including the composition of monadic data types such as lists, trees, exceptions and state.

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 Abstract bivariant Cuntz semigroups II

2020 ◽  
Vol 32 (1) ◽  
pp. 45-62 ◽  
Author(s):  
Ramon Antoine ◽  
Francesc Perera ◽  
Hannes Thiel
Keyword(s):  
Tensor Product ◽  
Monoidal Category ◽  
Additional Structure ◽  
Mild Conditions ◽  
Symmetric Monoidal Category

AbstractWe previously showed that abstract Cuntz semigroups form a closed symmetric monoidal category. This automatically provides additional structure in the category, such as a composition and an external tensor product, for which we give concrete constructions in order to be used in applications. We further analyze the structure of not necessarily commutative {\mathrm{Cu}}-semirings, and we obtain, under mild conditions, a new characterization of solid {\mathrm{Cu}}-semirings R by the condition that {R\cong\llbracket R,R\rrbracket}.

scholarly journals Tannakization in derived algebraic geometry

2014 ◽  
Vol 14 (3) ◽  
pp. 642-700 ◽  
Author(s):  
Isamu Iwanari
Keyword(s):  
Algebraic Geometry ◽  
Galois Group ◽  
Monoidal Category ◽  
Group Scheme ◽  
Affine Group ◽  
Stable Category ◽  
Group Schemes ◽  
Symmetric Monoidal Category ◽  
Derived Algebraic Geometry ◽  
Mixed Motives

AbstractIn this paper we begin studying tannakian constructions in ∞-categories and combine them with the theory of motivic categories developed by Hanamura, Levine, and Voevodsky. This paper is the first in a series of papers. For the purposes above, we first construct a derived affine group scheme and its representation category from a symmetric monoidal ∞-category, which we shall call the tannakization of a symmetric monoidal ∞-category. It can be viewed as an ∞-categorical generalization of work of Joyal-Street and Nori. Next we apply it to the stable ∞-category of mixed motives equipped with the realization functor of a mixed Weil cohomology. We construct a derived motivic Galois group which represents the automorphism group of the realization functor, and whose representation category satisfies an appropriate universal property. As a consequence, we construct an underived motivic Galois group of mixed motives, which is a pro-algebraic group and has nice properties. Also, we present basic properties of derived affine group schemes in the Appendix.

Hopf Quasimodules and Yetter-Drinfeld Modules over Hopf Quasigroups

2021 ◽  
Vol 28 (02) ◽  
pp. 213-242
Author(s):  
Tao Zhang ◽  
Yue Gu ◽  
Shuanhong Wang
Keyword(s):  
Monoidal Category ◽  
Drinfeld Modules ◽  
Braided Monoidal Category ◽  
Symmetric Monoidal Category ◽  
Category Equivalence ◽  
Hopf Modules ◽  
Hopf Quasigroup

We introduce the notions of a four-angle Hopf quasimodule and an adjoint quasiaction over a Hopf quasigroup [Formula: see text] in a symmetric monoidal category [Formula: see text]. If [Formula: see text] possesses an adjoint quasiaction, we show that symmetric Yetter-Drinfeld categories are trivial, and hence we obtain a braided monoidal category equivalence between the category of right Yetter-Drinfeld modules over [Formula: see text] and the category of four-angle Hopf modules over [Formula: see text] under some suitable conditions.

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 Abstract Bivariant Cuntz Semigroups

2018 ◽  
Vol 2020 (17) ◽  
pp. 5342-5386
Author(s):  
Ramon Antoine ◽  
Francesc Perera ◽  
Hannes Thiel
Keyword(s):  
Monoidal Category ◽  
Order Zero ◽  
Symmetric Monoidal Category ◽  
Cuntz Semigroup ◽  
Universal Coefficient Theorem

Abstract We show that abstract Cuntz semigroups form a closed symmetric monoidal category. Thus, given Cuntz semigroups $S$ and $T$, there is another Cuntz semigroup $[\![ S,T ]\!] $ playing the role of morphisms from $S$ to $T$. Applied to $C^{\ast }$-algebras $A$ and $B$, the semigroup $[\![ \operatorname{Cu}(A),\operatorname{Cu}(B) ]\!] $ should be considered as the target in analogs of the universal coefficient theorem for bivariant theories of Cuntz semigroups. Abstract bivariant Cuntz semigroups are computable in a number of interesting cases. We also show that order-zero maps between $C^{\ast }$-algebras naturally define elements in the respective bivariant Cuntz semigroup.

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