scholarly journals A quantum double construction in Rel

2012 ◽  
Vol 22 (4) ◽  
pp. 618-650 ◽  
Author(s):  
MASAHITO HASEGAWA
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Linear Logic ◽  
Binary Relations ◽  
Monoidal Categories ◽  
Quantum Double ◽  
Closed Category ◽  
Category Of Modules ◽  
Extra Structure

We study bialgebras and Hopf algebras in the compact closed categoryRelof sets and binary relations. Various monoidal categories with extra structure arise as the categories of (co)modules of bialgebras and Hopf algebras inRel. In particular, for any groupG, we derive a ribbon category of crossedG-sets as the category of modules of a Hopf algebra inRelthat is obtained by the quantum double construction. This category of crossedG-sets serves as a model of the braided variant of propositional linear logic.

Hopf algebras and linear logic

1996 ◽  
Vol 6 (2) ◽  
pp. 189-212 ◽  
Author(s):  
Richard F. Blute
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Linear Logic ◽  
Vector Spaces ◽  
Monoidal Categories ◽  
Simple Manner ◽  
Linear Topology ◽  
Topological Vector Spaces ◽  
Expository Paper ◽  
Involutive Negation

It has recently become evident that categories of representations of Hopf algebras provide fundamental examples of monoidal categories. In this expository paper, we examine such categories as models of (multiplicative) linear logic. By varying the Hopf algebra, it is possible to model several variants of linear logic. We present models of the original commutative logic, the noncommutative logic of Lambek and Abrusci, the braided variant due to the author, and the cyclic logic of Yetter. Hopf algebras provide a unifying framework for the analysis of these variants. While these categories are monoidal closed, they lack sufficient structure to model the involutive negation of classical linear logic. We recall work of Lefschetz and Barr in which vector spaces are endowed with an additional topological structure, called linear topology. The resulting category has a large class of reflexive objects, which form a *-autonomous category, and so model the involutive negation. We show that the monoidal closed structure of the category of representations of a Hopf algebra can be extended to this topological category in a natural and simple manner. The models we obtain have the advantage of being nondegenerate in the sense that the two multiplicative connectives, tensor and par, are not equated. It has been recently shown by Barr that this category of topological vector spaces can be viewed as a subcategory of a certain Chu category. In an Appendix, Barr uses this equivalence to analyze the structure of its tensor product.

The shuffle Hopf algebra and noncommutative full completeness

1998 ◽  
Vol 63 (4) ◽  
pp. 1413-1436 ◽  
Author(s):  
R. F. Blute ◽  
P. J. Scott
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Linear Logic ◽  
Natural Extension ◽  
Additive Group ◽  
Completeness Theorem ◽  
Vector Spaces ◽  
Invariance Condition ◽  
Shuffle Algebra ◽  
Topological Vector Spaces

AbstractWe present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic (CyLL). The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformations being equivariant under certain actions of a noncocommutative Hopf algebra called the shuffle algebra Multiplicative sequents are assigned a vector space of such dinaturals, and we show that this space has as a basis the denotations of cut-free proofs in CyLL + MIX. This can be viewed as a fully faithful representation of a free *-autonomous category, canonically enriched over vector spaces.This paper is a natural extension of the authors' previous work, “Linear Läuchli Semantics”, where a similar theorem is obtained for the commutative logic MLL + MIX. In that paper, we interpret proofs as dinaturals which are invariant under certain actions of the additive group of integers. Here we also present a simplification of that work by showing that the invariance criterion is actually a consequence of dinaturality. The passage from groups to Hopf algebras in this paper corresponds to the passage from commutative to noncommutative logic. However, in our noncommutative setting, one must still keep the invariance condition on dinaturals.

scholarly journals Comodule algebras and 2-cocycles over the (Braided) Drinfeld double

2019 ◽  
Vol 21 (04) ◽  
pp. 1850045 ◽  
Author(s):  
Robert Laugwitz
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Monoidal Category ◽  
Crossed Product ◽  
Braided Monoidal Category ◽  
Drinfeld Double ◽  
Comodule Algebra ◽  
Category Of Modules ◽  
Cohomology Spaces ◽  
Quasitriangular Hopf Algebra

We show that for dually paired bialgebras, every comodule algebra over one of the paired bialgebras gives a comodule algebra over their Drinfeld double via a crossed product construction. These constructions generalize to working with bialgebra objects in a braided monoidal category of modules over a quasitriangular Hopf algebra. Hence two ways to provide comodule algebras over the braided Drinfeld double (the double bosonization) are provided. Furthermore, a map of second Hopf algebra cohomology spaces is constructed. It takes a pair of 2-cocycles over dually paired Hopf algebras and produces a 2-cocycle over their Drinfeld double. This construction also has an analogue for braided Drinfeld doubles.

scholarly journals COQUASITRIANGULAR STRUCTURES FOR EXTENSIONS OF HOPF ALGEBRAS. APPLICATIONS

2012 ◽  
Vol 55 (1) ◽  
pp. 201-215 ◽  
Author(s):  
A. L. AGORE
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Quantum Double ◽  
Bijective Correspondence ◽  
Infinite Dimensional ◽  
Main Application ◽  
Split Extensions ◽  
Necessary And Sufficient

AbstractLet A ⊆ E be an extension of Hopf algebras such that there exists a normal left A-module coalgebra map π : E → A that splits the inclusion. We shall describe the set of all coquasitriangular structures on the Hopf algebra E in terms of the datum (A, E, π) as follows: first, any such extension E is isomorphic to a unified product A ⋉ H, for some unitary subcoalgebra H of E (A. L. Agore and G. Militaru, Unified products and split extensions of Hopf algebras, to appear in AMS Contemp. Math.). Then, as a main theorem, we establish a bijective correspondence between the set of all coquasitriangular structures on an arbitrary unified product A ⋉ H and a certain set of datum (p, τ, u, v) related to the components of the unified product. As the main application, we derive necessary and sufficient conditions for Majid's infinite-dimensional quantum double Dλ(A, H) = A ⋈τH to be a coquasitriangular Hopf algebra. Several examples are worked out in detail.

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.

COREPRESENTATION THEORY OF MULTIPLIER HOPF ALGEBRAS II

2000 ◽  
Vol 11 (02) ◽  
pp. 233-278 ◽  
Author(s):  
HIDEKI KUROSE ◽  
ALFONS VAN DAELE ◽  
YINHUO ZHANG
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Covariant Theory ◽  
Crossed Products ◽  
Quantum Double ◽  
Multiplier Algebra ◽  
Algebraic Quantum ◽  
Multiplier Hopf Algebras ◽  
Multiplier Hopf Algebra ◽  
Invertible Elements

We continue our development of the corepresentation theory of multiplier Hopf algebras. In this paper, we consider the corepresentations of a multiplier Hopf algebra A in a nondegenerate algebra B rather than on a vector space (cf. [25]). We concentrate ourself on those corepresentations of A in B which are invertible elements of the multiplier algebra M(B⊗A). They are called the unitary corepresentations of A. In particular, the generalized R-matrices or quasi-triangular structures of a regular multiplier Hopf algebra are unitary (bi)corepresentations. As an application the quantum double of an algebraic quantum group can be constructed by means of the universal unitary corepresentation. Moreover, a unitary corepresentation of A in B can implement an inner coaction of A on B which allows us to study the covariant theory and crossed products.

scholarly journals Separable Functors and Formal Smoothness

2007 ◽  
Vol 1 (3) ◽  
pp. 535-582 ◽  
Author(s):  
Alessandro Ardizzoni
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Algebra Homomorphism ◽  
Monoidal Categories ◽  
Invariant Integrals ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Algebra Theory ◽  
Formal Smoothness ◽  
Dual Case

AbstractThe natural problem we approach in the present paper is to show how the notion of formally smooth (co)algebra inside monoidal categories can substitute that of (co)separable (co)algebra in the study of splitting bialgebra homomorphisms. This is performed investigating the relation between formal smoothness and separability of certain functors and led to other results related to Hopf algebra theory. Between them we prove that the existence of ad-(co)invariant integrals for a Hopf algebra H is equivalent to the separability of some forgetful functors. In the finite dimensional case, this is also equivalent to the separability of the Drinfeld Double D(H) over H. Hopf algebras which are formally smooth as (co)algebras are characterized. We prove that if π : E → H is a bialgebra surjection with nilpotent kernel such that H is a Hopf algebra which is formally smooth as a K-algebra, then π has a section which is a right H-colinear algebra homomorphism. Moreover, if H is also endowed with an ad-invariant integral, then this section can be chosen to be H-bicolinear. We also deal with the dual case.

TWISTED DRINFELD DOUBLES AND REPRESENTATIONS OF A HOPF ALGEBRA

2012 ◽  
Vol 11 (06) ◽  
pp. 1250118
Author(s):  
LI BAI ◽  
SHUANHONG WANG
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Direct Proof ◽  
Irreducible Representations ◽  
Quantum Double ◽  
Drinfeld Double ◽  
Braided Group ◽  
R Matrix ◽  
Coalgebra Structure

In this paper, we consider a class of non-commutative and non-cocommutative Hopf algebras Hp(α, q, m) and then show that these Hopf algebras can be realized as a quantum double of certain Hopf algebras with respect to Hopf skew pairings (Ap(q, m), Bp(q, m), τα). Furthermore, using the Hopf skew pairing with appropriate group homomorphisms: ϕ : π → Aut (Ap(q, m)) and ψ : π → Aut (Bp(q, m)), we construct a twisted Drinfeld double D(Ap(q, m), Bp(q, m), τ; ϕ, ψ) which is a Turaev [Formula: see text]-coalgebra, where the group [Formula: see text] is a twisted semi-direct square of a group π. Then we obtain its quasi-triangular Turaev [Formula: see text]-coalgebra structure. We also study irreducible representations of Hp(1, q, m) and construct a corresponding R-matrix. Finally, we introduce the notion of a left Yetter–Drinfeld category over a Turaev group coalgebra and show that such a category is a Turaev braided group category by a direct proof, without center construction. As an application, we consider the case of the quasi-triangular Turaev [Formula: see text]-coalgebra structure on our twisted Drinfeld double.

scholarly journals Cartesian Differential Categories as Skew Enriched Categories

2021 ◽  
Author(s):  
Richard Garner ◽  
Jean-Simon Pacaud Lemay
Keyword(s):  
Linear Logic ◽  
Usual Sense ◽  
Enriched Category ◽  
Enriched Categories ◽  
Full Structure ◽  
Monoidal Closed Category ◽  
Category Of Modules ◽  
Differential Categories ◽  
Linear Category ◽  
Open Question

AbstractWe exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind of enriched category. The base for the enrichment is the category of commutative monoids—or in a straightforward generalisation, the category of modules over a commutative rig k. However, the tensor product on this category is not the usual one, but rather a warping of it by a certain monoidal comonad Q. Thus the enrichment base is not a monoidal category in the usual sense, but rather a skew monoidal category in the sense of Szlachányi. Our first main result is that cartesian differential categories are the same as categories with finite products enriched over this skew monoidal base. The comonad Q involved is, in fact, an example of a differential modality. Differential modalities are a kind of comonad on a symmetric monoidal k-linear category with the characteristic feature that their co-Kleisli categories are cartesian differential categories. Using our first main result, we are able to prove our second one: that every small cartesian differential category admits a full, structure-preserving embedding into the cartesian differential category induced by a differential modality (in fact, a monoidal differential modality on a monoidal closed category—thus, a model of intuitionistic differential linear logic). This resolves an important open question in this area.

Double-bosonization of braided groups and the construction of Uq(g)

1999 ◽  
Vol 125 (1) ◽  
pp. 151-192 ◽  
Author(s):  
S. MAJID
Keyword(s):  
Hopf Algebra ◽  
Quantum Groups ◽  
Cartan Subalgebra ◽  
Positive Root ◽  
Point Of View ◽  
Root Space ◽  
Quantum Double ◽  
Braided Group ◽  
Negative Root ◽  
Quasitriangular Hopf Algebra

We introduce a quasitriangular Hopf algebra or ‘quantum group’ U(B), the double-bosonization, associated to every braided group B in the category of H-modules over a quasitriangular Hopf algebra H, such that B appears as the ‘positive root space’, H as the ‘Cartan subalgebra’ and the dual braided group B* as the ‘negative root space’ of U(B). The choice B=Uq(n+) recovers Lusztig's construction of Uq(g); other choices give more novel quantum groups. As an application, our construction provides a canonical way of building up quantum groups from smaller ones by repeatedly extending their positive and negative root spaces by linear braided groups; we explicitly construct Uq(sl3) from Uq(sl2) by this method, extending it by the quantum-braided plane. We provide a fundamental representation of U(B) in B. A projection from the quantum double, a theory of double biproducts and a Tannaka–Krein reconstruction point of view are also provided.

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