弱Hopf代数上的扭积

在弱Hopf代数上,定义了交叉积概念,并且得到了它的两种特殊形式,冲积和扭积．特别地,给出了扭积为弱Hopf代数的一个充要条件,推广了Hopf代数的相应结论．     

On Quasi-Bicrossed Products of Weak Hopf Algebras

The aim of this paper is to study quasi-bicrossed products and a especially quantum quasi-doubles. Firstly, we construct one new kind of quasi-bicrossed products by weak Hopf algebras and then devote a brief discussion to this matter. And, we discuss the conditions for quasi-bicrossed products to possess the structure of almost weak Hopf algebras, containing the case of a special smash product. At the end, we give some properties on the quantum quasi-double, respectively on the quasi-R-isomorphism, the representation-theoretic interpretation and the regularity of the quasi-R-matrix.     

Tensor Product of Massey Products

In this paper, we interpret Massey products in terms of realizations （twitsting cochains） of certain differential graded coalgebras with values in differential graded algebras. In the case where the target algebra is the cobar construction of a differential graded commutative Hopf algebra, we construct the tensor product of realizations and show that the tensor product is strictly associative, and commutative up to homotopy.     

弱Hopf代数作用与冲积

本文研究了弱Hopf代数上的冲积并讨论了它约性质．设H是弱Hopf代数,A是左H-摸代数．我们给出了冲积A#H是弱双代数的一个充分条件以及A#H是A可分扩张的一个判定条件．另外,利用积分理论研究了Hopf模代数的有限性条件．     

Projective Summands in Tensor Products of Simple Modules of Finite Dimensional Hopf Algebras

Abstract

Let H be a finite dimensional Hopf algebra over a field k. We show that H contains a unique maximal Hopf ideal J _w (H) contained in J(H), the Jacobson radical of H. We give various characterizations of J _w (H), for example J _w (H) = Ann _H ((H/J(H))^?n ) for all large enough n. The smallest positive integer n with this property is denoted by l _w (H). We prove that l _w (H) equals the smallest number n such that (H/J(H))^?n contains every projective indecomposable H/J _w (H)-module as a direct summand. This also equals the minimal n such that the tensor product of n suitable simple H-modules contains the projective cover of the trivial H/J _w (H)-module as a direct summand. We define projective homomorphisms between H-modules, which are used to obtain various reciprocity laws for tensor products of simple H-modules and their projective indecomposable direct summands. We also discuss some consequences of our general results in case H = kG is a group algebra of a finite group G and k is a field of characteristic p.     

Semiartinian modules relative to a torsion theory

In the first section we generalize the concept of the socle of a module by replacing simples with τ-simple modules for a hereditary torsion theory τ. The second section is concerned with the τ-Loewy series, and finally these general results are applied in the section 3 to the notions of τ-semiartinian rings and modules.     

One-sided Hopf algebras and quantum quasigroups

We study the connections between one-sided Hopf algebras and one-sided quantum quasigroups, tracking the four possible invertibility conditions for the left and right composite morphisms that combine comultiplications and multiplications in these structures. The genuinely one-sided structures exhibit precisely two of the invertibilities, while it emerges that imposing one more condition often entails the validity of all four. A main result shows that under appropriate conditions, just one of the invertibility conditions is su?cient for the existence of a one-sided antipode. In the left Hopf algebra which is a variant of the quantum special linear group of two-dimensional matrices, it is shown explicitly that the right composite is not injective, and the left composite is not surjective.     

Hopf代数的结构定理和对映阶数

本文中,我们把Ｈｏｐｆ代数的结构定理推广到Ｈｏｐｆ代数意义下的同构,从而给出Ｈｏｐｆ代数既约分支的对映阶数,并得到Ｈｏｐｆ代数扩张的对映阶数是任意的．这部分回答了Ｅ．Ｊ．Ｔａｆｔ１９９４年提出的一个问题．     

Extension of a quantized enveloping algebra by a Hopf algebra

Suppose that H is a Hopf algebra,and g is a generalized Kac-Moody algebra with Cartan matrix A =(aij)I×I,where I is an index set and is equal to either {1,2,...,n} or the natural number set N.Let f,g be two mappings from I to G(H),the set of group-like elements of H,such that the multiplication of elements in the set {f(i),g(i)|i ∈I} is commutative.Then we define a Hopf algebra Hgf Uq(g),where Uq(g) is the quantized enveloping algebra of g.     

Hochschild (Co)Homology of Hopf Crossed Products*

For a general crossed product E = A#_f H, of an algebra A by a Hopf algebra H, we obtain complexes smaller than the canonical ones, giving the Hochschild homology and cohomology of E. These complexes are equipped with natural filtrations. The spectral sequences associated to them coincide with the ones obtained using a natural generalization of the direct method introduced in Trans. Amer. Math. Soc. 74 (1953) 110–134. We also get that if the 2-cocycle f takes its values in a separable subalgebra of A, then the Hochschild (co)homology of E with coefficients in M is the (co)homology of H with coefficients in a (co)chain complex.     

Curves on Quasi-Schemes

This paper concerns curves on noncommutative schemes, hereafter called quasi-schemes. Aquasi-scheme X is identified with the category ofquasi-coherent sheaves on it. Let X be a quasi-scheme having a regularly embeddedhypersurface Y. Let C be a curve on X which is in good position withrespect to Y (see Definition 5.1) – this definition includes a requirement that Xbe far from commutative in a certain sense. Then C is isomorphic to , where n is the number of points of intersection of Cwith Y. Here , or rather , is the quotient category -graded modules over the commutative polynomial ring, modulo the subcategory ofmodules having Krull dimension n – 2. This is a hereditary category whichbehaves rather like , the category of quasi-coherentsheaves on .     

On Semisimple Hopf Algebras of Dimension 2 m

In this paper we classify all nontrivial semisimple Hopf algebras of dimension 2 ^{n +1} with the group of grouplikes isomorphic to _{2 n–1}×₂. Moreover, we extend some results on irreducible representations from groups to semisimple Hopf algebras and prove that certain semisimple Hopf algebras, including the ones classified in this paper, satisfy the generalized power map property.     

W-代数W(2,2)的单参数量子形变(英文)

众所周知,泛包络代数的量子形变所对应的量子群结构是依据所给代数的单根系给出的.我们构造了q-形变W代数Wq,并给出其非平凡的量子群结构.     

交叉余积上的Hopf代数结构

本文给出了Ｓｍａｓｈ积代数结构和交叉余积余代数结构构成双代数的一个充分必要条件．另外,还给出了这一新的双代数成为Ｈｏｐｆ代数的一个充分条件．     

Symmetric Designs Attached to Four-Weight Spin Models

H. Guo and T. Huang studied the four-weight spin models (X, W ₁, W ₂, W ₃, W ₄;D) with the property that the entries of the matrix W ₂ (or equivalently W ₄) consist of exactly two distinct values. They found that such spin models are always related to symmetric designs whose derived design with respect to any block is a quasi symmetric design. In this paper we show that such a symmetric design admits a four-weight spin model with exactly two values on W ₂ if and only if it has some kind of duality between the set of points and the set of blocks. We also give some examples of parameters of symmetric designs which possibly admit four-weight spin models with exactly two values on W ₂.