Pairing and Quantum Double of Multiplier Hopf Algebras

We define and investigate pairings of multiplier Hopf (*-)algebras which are nonunital generalizations of Hopf algebras. Dual pairs of multiplier Hopf algebras arise naturally from any multiplier Hopf algebra A with integral and its dual Â. Pairings of multiplier Hopf algebras play a basic rôle, e.g., in the study of actions and coactions, and, in particular, in the relation between them. This aspect of the theory is treated elsewhere. In this paper we consider the quantum double construction out of a dual pair of multiplier Hopf algebras. We show that two dually paired regular multiplier Hopf (*-)algebras A and B yield a quantum double which is again a regular multiplier Hopf (*-)algebra. If A and B have integrals, then the quantum double also has an integral. If A and B are Hopf algebras, then the quantum double multiplier Hopf algebra is the usual quantum double. The quantum double construction for dually paired multiplier Hopf (*-)algebras yields new nontrivial examples of multiplier Hopf (*-)algebras.     

Crossed modules,quantum braided groups,and ribbon structures

Previous results about crossed modules over a braided Hopf algebra are applied to the study of quantum groups in braided categories. Cross products for braided Hopf algebras and quantum braided groups are constructed. Criteria for when a braided Hopf algebra or a quantum group is a cross product are obtained. A generalization of Majid's transmutation procedure for quantum braided groups is considered. A ribbon structure on a quantum braided group and its compatibility with cross product and transmutation are studied.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 3, pp. 368–387, June, 1995.     

Extensions of Finite Quantum Groups by Finite Groups

We give a necessary and sufficient condition for two Hopf algebras presented as central extensions to be isomorphic, in a suitable setting. We then study the question of isomorphism between the Hopf algebras constructed in [AG] as quantum subgroups of quantum groups at roots of 1. Finally, we apply the first general result to show the existence of infinitely many non-isomorphic Hopf algebras of the same dimension, presented as extensions of finite quantum groups by finite groups. Partially supported by CONICET, ANPCyT, Secyt (UNC) and Ministerio de Ciencia y Tecnología de la Provincia de Córdoba.     

Algebraic properties of quantum quasigroups

Quantum quasigroups provide a self-dual framework for the unification of quasigroups and Hopf algebras. This paper furthers the transfer program, investigating extensions to quantum quasigroups of various algebraic features of quasigroups and Hopf algebras. Part of the difficulty of the transfer program is the fact that there is no standard model-theoretic procedure for accommodating the coalgebraic aspects of quantum quasigroups. The linear quantum quasigroups, which live in categories of modules under the direct sum, are a notable exception. They form one of the central themes of the paper.From the theory of Hopf algebras, we transfer the study of grouplike and setlike elements, which form separate concepts in quantum quasigroups. From quasigroups, we transfer the study of conjugate quasigroups, which reflect the triality symmetry of the language of quasigroups. In particular, we construct conjugates of cocommutative Hopf algebras. Semisymmetry, Mendelsohn, and distributivity properties are formulated for quantum quasigroups. We classify distributive linear quantum quasigroups that furnish solutions to the quantum Yang-Baxter equation. The transfer of semisymmetry is designed to prepare for a quantization of web geometry.     

弱准三角Hopf代数和解量子Yang-Baxter方程

出于解量子Yang-Baxter方程的需要,本文定义了弱准三角Hopf代数,并且发现了一类构造弱准三角Hopf代数的方法,文中称之为杨-积,它可以提供量子Yang-Baxter方程的解.     

Coorbits of Quantum Homogeneous Spaces

The notion of coorbits for spaces with quantum group actions is introduced. A space with a quantum group action is given by a pair of algebras: an associative algebra which is the analog of a classical topological space, and a Hopf algebra which is the analog of a classical topological group. The Hopf algebra acts on the associative algebra via a comodule structure mapping which is also an algebra homomorphism. For a space with a quantum group action, a coorbit is a pair of spaces given by the image and the kernel of an algebra homomorphism from the associative algebra to the Hopf algebra. The coorbits of several types of quantum homogeneous spaces are discussed. In the case when the associative algebra is the group algebra of a group and the Hopf algebra is a quotient of the group algebra, the connection between the set of coorbits and the character group is established.     

Combinatorial Hopf algebras from renormalization

In this paper we describe the right-sided combinatorial Hopf structure of three Hopf algebras appearing in the context of renormalization in quantum field theory: the non-commutative version of the Faà di Bruno Hopf algebra, the non-commutative version of the charge renormalization Hopf algebra on planar binary trees for quantum electrodynamics, and the non-commutative version of the Pinter renormalization Hopf algebra on any bosonic field.     

A Two-Parameter Quantum (2+1)-Superspace and its Deformed Derivation Algebra as Hopf Superalgebra

As is well known, a Hopf algebra setting is an efficient tool to study some geometric structures such as the Maurer-Cartan invariant forms and the corresponding vector fields on a noncommutative space. In this study we introduce a two-parameter quantum (2+1)-superspace with a Hopf superalgebra structure.We also define some derivation operators acting on this quantum superspace, and we show that the algebra of these derivations is a Hopf superalgebra. Furthermore it will be shown how the derivation operators lead to a bicovariant differential calculus on the two- parameter quantum (2+1)-superspace. In conclusion, based on the bicovariant differential calculus, the Maurer-Cartan right invariant differential forms and the corresponding quantum Lie superalgebra are given.     

弱Hopf群T-余代数上的弱Doi-Hopf群模

在弱Hopf群T-余代数情形下,弱量子Yetter-Drinfeld群模的概念被引入,并证明了弱量子Yetter-Drinfeld群模是特殊的弱Doi-Hopf群模.接着建立了弱量子Yetter Drinfeld群模范畴与弱Hopf群双余模代数的余不动点子代数B上模范畴之间的伴随对.最后考虑了弱量子Yetter-Drinfeld群模的积分.     

The trace space and Kauffman's knot invariants

The traces in the construction of Kauffman's knot invariants are studied. The trace space is determined for a semisimple finite-dimensional quantum Hopf algebra and the best lower bound of the dimension of the trace space is given for a unimodular finite-dimensional quantum Hopf algebra.

     

弱Hopf量子Yang-Baxter H-模基本定理(英文)

本文引入弱Hopf量子Yang-Baxter模概念.利用弱Hopf模基本定理的方法,获得了弱Hopf量子Yang-Baxter模基本定理,进一步还得到了相关Hopf模基本定理.     

拟三角Hopf代数的Ore -扩张

该文主要考虑了拟三角Hopf代数的某种Ore -扩张问题. 对拟三角Hopf代数的Ore -扩张何时保持相同的拟三角结构给出了充分必要条件. 最后作为应用, 文章讨论了Sweedler Hopf代数和Lusztig小量子群的Ore -扩张结构.     

Graded Quantum Groups and Quasitriangular Hopf Group-Coalgebras

ABSTRACT

Starting from a Hopf algebra endowed with an action of a group π by Hopf automorphisms, we construct (by a “twisted” double method) a quasitriangular Hopf π-coalgebra. This method allows us to obtain non-trivial examples of quasitriangular Hopf π-coalgebras for any finite group π and for infinite groups π such as GL _n (𝕂). In particular, we define the graded quantum groups, which are Hopf π-coalgebras for π = ?[[h]] ^l and generalize the Drinfeld-Jimbo quantum enveloping algebras.     

Quantum algebras,quantum coalgebras,invariants of 1-1 tangles and knots

Quantum coalgebras are defined and studied. A theory of asso­ciated invariants of 1-1 tangles, knots and links is developed. The notion of quantum coalgebra is more general than dual of quantum algebra. Examples of quantum algebras include quasitriangular Hopf algebras and examples of quantum coalgebras include coquasi triangu­lar Hopf algebras.     

Cibils–Rosso's Theorem for Quantum Groupoids

Abstract

We generalize the Cibils–Rosso's theorem for categories of Sweedler's Hopf bimodules to the one for categories of weak entwined bimodules. We show that the weak entwined bimodules are modules over a certain algebra. Our best results are attained for categories of weak Hopf bimodules over quantum groupoids (weak Hopf algebras), as special cases of weak Doi–Hopf bimodules.     

An Algebraic Framework for Group Duality

A Hopf algebra is a pair (A, Δ) whereAis an associative algebra with identity andΔa homomorphism formAtoA⊗Asatisfying certain conditions. If we drop the assumption thatAhas an identity and if we allowΔto have values in the so-called multiplier algebraM(A⊗A), we get a natural extension of the notion of a Hopf algebra. We call this a multiplier Hopf algebra. The motivating example is the algebra of complex functions with finite support on a group with the comultiplication defined as dual to the product in the group. Also for these multiplier Hopf algebras, there is a natural notion of left and right invariance for linear functionals (called integrals in Hopf algebra theory). We show that, if such invariant functionals exist, they are unique (up to a scalar) and faithful. For a regular multiplier Hopf algebra (A, Δ) (i.e., with invertible antipode) with invariant functionals, we construct, in a canonical way, the dual (Â, Δ). It is again a regular multiplier Hopf algebra with invariant functionals. It is also shown that the dual of (Â, Δ) is canonically isomorphic with the original multiplier Hopf algebra (A, Δ). It is possible to generalize many aspects of abstract harmonic analysis here. One can define the Fourier transform; one can prove Plancherel's formula. Because any finite-dimensional Hopf algebra is a regular multiplier Hopf algebra and has invariant functionals, our duality theorem applies to all finite-dimensional Hopf algebras. Then it coincides with the usual duality for such Hopf algebras. But our category of multiplier Hopf algebras also includes, in a certain way, the discrete (quantum) groups and the compact (quantum) groups. Our duality includes the duality between discrete quantum groups and compact quantum groups. In particular, it includes the duality between compact abelian groups and discrete abelian groups. One of the nice features of our theory is that we have an extension of this duality to the non-abelian case, but within one category. This is shown in the last section of our paper where we introduce the algebras of compact type and the algebras of discrete type. We prove that also these are dual to each other. We treat an example that is sufficiently general to illustrate most of the different features of our theory. It is also possible to construct the quantum double of Drinfel'd within this category. This provides a still wider class of examples. So, we obtain many more than just the compact and discrete quantum within this setting.     

Quantum families of quantum group homomorphisms

The notion of a quantum family of maps has been introduced in the framework of C*-algebras. As in the classical case, one may consider a quantum family of maps preserving additional structures (e.g. quantum family of maps preserving a state). In this paper, we define a quantum family of homomorphisms of locally compact quantum groups. Roughly speaking, we show that such a family is classical. The purely algebraic counterpart of the discussed notion, i.e. a quantum family of homomorphisms of Hopf algebras, is introduced and the algebraic counterpart of the aforementioned result is proved. Moreover, we show that a quantum family of homomorphisms of Hopf algebras is consistent with the counits and coinverses of the given Hopf algebras. We compare our concept with weak coactions introduced by Andruskiewitsch and we apply it to the analysis of adjoint coaction.     

Representation-theoretic rank and double hopf algebras

We compute the intrinsic category-theoretic rank: for quasitriangular Hopf algebras in the case of the quantum double Hopf algebra of Drinfeld. The result is closely related ti recent Hopf algebra invariants of Larson and Radford.     

A New Cyclic Module for Hopf Algebras

We define a new cyclic module, dual to the Connes–Moscovici cocyclic module, for Hopf algebras, and give a characteristic map for coactions of Hopf algebras. We also compute the resulting cyclic homology for cocommutative Hopf algebras, and some quantum groups.     

Hopf algebras of primitive Lie pseudogroups and Hopf cyclic cohomology

We associate to each infinite primitive Lie pseudogroup a Hopf algebra of ‘transverse symmetries,’ by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a ‘quantum group’ counterpart of the infinite-dimensional primitive Lie algebra of the pseudogroup. It is first constructed via its action on the étale groupoid associated to the pseudogroup, and then realized as a bicrossed product of a universal enveloping algebra by a Hopf algebra of regular functions on a formal group. The bicrossed product structure allows to express its Hopf cyclic cohomology in terms of a bicocyclic bicomplex analogous to the Chevalley-Eilenberg complex. As an application, we compute the relative Hopf cyclic cohomology modulo the linear isotropy for the Hopf algebra of the general pseudogroup, and find explicit cocycle representatives for the universal Chern classes in Hopf cyclic cohomology. As another application, we determine all Hopf cyclic cohomology groups for the Hopf algebra associated to the pseudogroup of local diffeomorphisms of the line.