Quantum Yang-Baxter module algebras

LetH be a quantum group over a commutative ringR. We introduce the concept of quantum Yang-BaxterH-module algebra, generalizing the notion ofH-dimodule algebra in the case whereH is commutative, cocommutative and faithfully projective. After discussing some examples, we introduceH-Azumaya algebras. The set of quivalence classes ofH-Azumaya algebras can be made into a group, called the Brauer group of the quantum groupH. This group is a generalization of the Brauer-Long group.This author wishes to thank the Department of Mathematics, UIA, for its hospitality and financial support during the time when most of this paper was written.     

The classical Yang-Baxter equation on alternative algebras: The alternative D-bialgebra structure on Cayley-Dickson matrix algebras

We consider the Yang-Baxter equations on alternative algebras and prove that the bialgebras induced by the solutions to these equations are alternative D-bialgebras. We describe the alternative D-bialgebra structure on Cayley-Dickson matrix algebras.     

Generalized q-Schur algebras and quantum Frobenius

The quantum Frobenius map and it splitting are shown to descend to maps between generalized q-Schur algebras at a root of unity. We also define analogs of q-Schur algebras for any affine algebra, and prove the corresponding results for them.     

A class of solutions to the quantum colored Yang-Baxter equation

In this paper, we give a new method to solve the quantum colored Yang-Baxter matrix equation (QCYBE), and a class of solutions to the QCYBE is given.     

Double Frobenius algebras

Some equivalent conditions for double Frobenius algebras to be strict ones are given. Then some examples of (strict or non-strict) double Frobenius algebras are presented. Finally, a sufficient and necessary condition for the trivial extension of a double Frobenius algebra to be a (strict) double Frobenius algebra is given.     

Frobenius Poisson algebras

This paper is devoted to study Frobenius Poisson algebras. We introduce pseudo-unimodular Poisson algebras by generalizing unimodular Poisson algebras, and investigate Batalin-Vilkovisky structures on their cohomology algebras. For any Frobenius Poisson algebra, all Eatalin-Vilkovisky opera tors on its Poisson cochain complex are described explicitly. It is proved that there exists a Batalin-Vilkovisky operator on its cohomology algebra which is induced from a Batalin-Vilkovisky operator on the Poisson cochain complex, if and only if the Poisson st rue ture is pseudo-unimodular. The relation bet ween modular derivations of polynomial Poisson algebras and those of their truncated Poisson algebras is also described in some cases.     

General solution of a kind of quantum coloured Yang-Baxter equation (I)

In this paper, we give a new method to solve the quantum coloured Yang-Baxter matrix equation (QCYBE), and the general solution for a kind of QCYBE is given.     

Frobenius induction for algebras

Let B→A be a homomorphism of Hopf algebras and let C be an algebra. We consider the induction from B to A of C in two cases: when C is a B-interior algebra and when C is a B-module algebra. Our main results establish the connection between the two inductions. The inspiration comes from finite group representation theory, and some constructions work in even more general contexts.     

Yang-Baxter方程的解与量子Yang-Baxter H-余模

设Ｈ是域ｋ上的Ｈｏｐｆ代数。本文首先讨论了量子Ｙａｎｇ－Ｂａｘｔｅｒ Ｈ－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。     

Super solutions of the dynamical Yang-Baxter equation

Solutions of the classical dynamical Yang-Baxter equation on a Lie superalgebra are called super dynamical matrices. A super dynamical matrix satisfies the zero weight condition if

    for all 

In this paper we classify super dynamical matrices with zero weight.

     

On quantum stabilizer codes derived from local Frobenius rings

In this paper we consider stabilizer codes over local Frobenius rings. Firstly, we study the relative minimum distances of a stabilizer code and its reduction onto the residue field. We show that for various scenarios, a free stabilizer code over the ring does not underperform the according stabilizer code over the field. This leads us to conjecture that the same is true for all free stabilizer codes. Secondly, we focus on the isometries of stabilizer codes. We present some preliminary results and introduce some interesting open problems.     

An algorithmic construction of group automorphisms and the quantum Yang-Baxter equation

The structure group G of a non-degenerate symmetric set (X,S) is a Bieberbach and a Garside group. We describe a combinatorial method to compute explicitly a group of automorphisms of G and show this group admits a subgroup that preserves the Garside structure. In some special cases, we could also prove the group of automorphisms found is an outer automorphism group.     

Frobenius morphisms and stable module categories of repetitive algebras

Let k be the algebraic closure of a finite field F_q and A be a finite dimensional k-algebra with a Frobenius morphism F.In the present paper we establish a relation between the stable module category of the repetitive algebra  of A and that of the repetitive algebra of the fixed-point algebra A~F.As an application,it is shown that the derived category of A~F is equivalent to the subcategory of F-stable objects in the derived category of A when A has a finite global dimension.     

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。     

The Structure of Frobenius Algebras and Separable Algebras

We present a unified approach to the study of separable and Frobenius algebras. The crucial observation is that both types of algebras are related to the nonlinear equation R¹²R²³=R²³R¹³=R¹³R¹², called the FS-equation. Solutions of the FS-equation automatically satisfy the braid equation, an equation that is in a sense equivalent to the quantum Yang–Baxter equation. Given a solution to the FS-equation satisfying a certain normalizing condition, we can construct a Frobenius algebra or a separable algebra A(R) – the normalizing condition is different in both cases. The main result of this paper is the structure of these two fundamental types of algebras: a finite dimensional Frobenius or separable k-algebra A is isomorphic to such an A(R). A(R) can be described using generators and relations. A new characterization of Frobenius extensions is given: B A is Frobenius if and only if A has a B-coring structure (A, , ) such that the comultiplication : A A_B A is an A-bimodule map.     

余半单Hopf代数的Frobenius性质

设k是特征为0的代数闭域,H为其上的余半单Hopf代数,本文证明了当H有型:l:1 m:p 1:q(其中p~2相似文献   

三维幂零李超代数的Yang-Baxter算子

在复数域C上,利用三维幂零李超代数的分类,通过计算刻画了三维幂零李超代数的Yang-Baxter算子.     

Compatible Lie Brackets and the Yang-Baxter Equation

We show that any pair of compatible Lie brackets with a common invariant form produces a nonconstant solution of the classical Yang-Baxter equation. We describe the corresponding Poisson brackets, Manin triples, and Lie bialgebras. It turns out that all bialgebras associated with the solutions found by Belavin and Drinfeld are isomorphic to some bialgebras generated by our solutions. For any compatible pair, we construct a double with a common invariant form and find the corresponding solution of the quantum Yang-Baxter equation for this double. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 2, pp. 195–207, February, 2006.