Yanglg-Baxter方程的解与量子Yang一BaxterH一余模(英)

设H是域k上的Hopf代数.本文首先讨论了量子Yang-BaxterH-余模与Yang-Baxter方程的解的关系;然后作为应用,给出了任意Hopf代数上Yang-Baxter方程的一个解.     

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.     

关于(f,T)-相容对(B,H)和_H~HYD中的Hopf代数

本文定义了(f,T)-相容对(B,H),利用这样的相容对可以给出一个辫子张量 范畴和一个量子Yang-Baxter方程的解,并且通过扭曲Hopf代数B的乘法,构造 Yetter-Drinfeld范畴中HHyD的Hopf代数.     

关于(f,т)-相容Hopf代数对(B,H)

该文定义了(f,τ)-相容Hopf代数对(B,H),利用这样的对(B,H),给出了左H-余模范畴HM的一个辫子张量子范畴,从而得到一个量子Yang-Baxter算子,并且通过扭曲Hopf代数B的乘法,构造出Yetter-Drinfeld范畴中H HYD的Hopf代数．     

关于(f,τ) -相容Hopf代数对(B,H)

该文定义了(f,τ) -相容Hopf代数对(B,H),利用这样的对(B,H),给出了左H -余模范畴^HM的一个辫子张量子范畴,从而得到一个量子Yang-Baxter算子,并且通过扭曲Hopf代数$B$的乘法,构造出Yetter-Drinfeld范畴中^H_HYD的Hopf代数.     

关于拟三角Hopf代数的Cylinder余代数和Cylinder余积(英)

本文引入两个概念,即,关于拟三角双代数的cylinder余代数和cylinder余积,并指出存在一个反余代数同构：(H,■)≌(H,■),其中(H,■)是cylinder余积,(H,■)是辫余积,对任意有限维Hopf代数H,我们证明Drinfel'd量子偶(D(H),■_(D(H)))是cylinder余积．设(H,H,R)是余配对Hopf代数,如果R∈Z(H■H),则通过两次扭曲,我们可以构造扭曲余代数(H~■)R~(-1),它的余乘法恰是cylinder余积．而且对任意的广义Long重模,通过cylinder扭曲,我们可以构造Yang-Baxter方程,四辫对和Long方程．     

半拟三角弱Hopf代数

作为拟三角弱Hopf代数的推广,我们引入了半拟三角弱Hopf代数的概念.令(H,R,v)是一个半拟三角弱Hopf代数,其中,R是其半拟三角结构.我们指明R保持了拟三角弱Hopf代数中泛R-矩阵的许多基本性质.特别地,讨论了Drinfeld元的性质,证明其是可逆的并且是余作用v的余不变量.另外,证明了半拟三角弱Hopf代数的对极平方是对合的.     

线状李超代数Ln,m上的Yang-Baxter 方程

在特征零的代数闭域上,首先做出Ln,m 的一个空间的直和分解,从而将Ln,m 上的Yang-Baxter 方程的解分为若干情形。然后分别在每种情形下对Yang-Baxter 方程进行求解,进而得到了Ln,m 上的所有的Yang-Baxter方程的解的矩阵形式。     

线状李超代数L_(n,m)上的Yang-Baxter方方程

在特征零的代数闭域上,首先做出L_(n,m)的一个空间的直和分解,从而将L_(n,m)上的Yang-Baxter方程的解分为若干情形.然后分别在每种情形下对Yang-Baxter方程进行求解,进而得到了L_(n,m)上的所有的Yang-Baxter方程的解的矩阵形式.     

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.     

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.     

Smash coproduct and braided groups in braided monoidal category

A smash coproduct in braided monoidal category C is constructed and some conditions making the smash coproduct a Hopf algebra or braided Hopf algebra are given. It is shown that the smash coproductB ×H in ^HM is equivalent to the transmutation of Hopf algebra. Thus a method for transmutation theory is provided. Let σ be 2-co-cycle andH a commutation Hopf algebra. A Hopf algebraHσ is constructed.Hσ?Hσ whereHσ is a transmutation ofHσ. The braided groups from some solutions of quantum Yang-Baxter equation are obtained.     

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.     

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.     

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.

     

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.     

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.     

Yang-Baxter algebra and generation of quantum integrable models

We discover an operator-deformed quantum algebra using the quantum Yang-Baxter equation with the trigonometric R-matrix. This novel Hopf algebra together with its q→1 limit seems the most general Yang-Baxter algebra underlying quantum integrable systems. We identify three different directions for applying this algebra in integrable systems depending on different sets of values of the deforming operators. Fixed values on the whole lattice yield subalgebras linked to standard quantum integrable models, and the associated Lax operators generate and classify them in a unified way. Variable values yield a new series of quantum integrable inhomogeneous models. Fixed but different values at different lattice sites can produce a novel class of integrable hybrid models including integrable matter-radiation models and quantum field models with defects, in particular, a new quantum integrable sine-Gordon model with defect. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 470–485, June, 2007.     

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.     

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.