Representations of Rank 3 Algebras

The class of rank 3 algebras includes the Jordan algebra of a symmetric bilinear form, the trace zero elements of a Jordan algebra of degree 3, pseudo-composition algebras, certain algebras that arise in the study of Riccati differential equations, as well as many other algebras. We investigate the representations of rank 3 algebras and show under some conditions on the eigenspaces of the left multiplication operator determined by an idempotent element that the finite-dimensional irreducible representations are all one-dimensional.     

A monoidal approach to splitting morphisms of bialgebras

The main goal of this paper is to investigate the structure of Hopf algebras with the property that either its Jacobson radical is a Hopf ideal or its coradical is a subalgebra. Let us consider a Hopf algebra such that its Jacobson radical is a nilpotent Hopf ideal and is a semisimple algebra. We prove that the canonical projection of on has a section which is an -colinear algebra map. Furthermore, if is cosemisimple too, then we can choose this section to be an -bicolinear algebra morphism. This fact allows us to describe as a `generalized bosonization' of a certain algebra in the category of Yetter-Drinfeld modules over . As an application we give a categorical proof of Radford's result about Hopf algebras with projections. We also consider the dual situation. Let be a bialgebra such that its coradical is a Hopf sub-bialgebra with antipode. Then there is a retraction of the canonical injection of into which is an -linear coalgebra morphism. Furthermore, if is semisimple too, then we can choose this retraction to be an -bilinear coalgebra morphism. Then, also in this case, we can describe as a `generalized bosonization' of a certain coalgebra in the category of Yetter-Drinfeld modules over .

     

关于余模余代数的L-R smash余积和L-R扭曲余积

由一种新方法给出了L-R smash余积的Mashke定理,并研究了L-R扭曲余积与左(右)扭曲偶的关系.     

The Relative Properties of Graded Ring R and Smash Product R # G

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The Duality Theorem for Weak Hopf Algebra (Co) Actions

In this article, we mainly study a new notion of a generalized smash product for weak Hopf comodule algebras and provide a new version of the duality theorem for weak smash products as an application.     

Fundamental Theorems of Weak Doi–Hopf Modules and Semisimple Weak Smash Product Hopf Algebras

Abstract

This paper, mainly gives a Fundamental Theorem of weak Doi–Hopf modules, which is not only generalizes the Fundamental Theorem of weak Hopf modules but also generalizes the Fundamental Theorem of relative Hopf modules. Moreover, it gives a sufficient and necessary condition for weak smash product algebras to be weak bialgebras, and a sufficient condition for weak smash product algebras to be semisimple weak Hopf algebras.     

H-separable Hopf Galois Extensions and Azumaya Algebra

1 IntroductionLet A/R be a ring extension with the common identity 1. A/R is said to be separable if theA-bimodule homomorphism of A @R A onto A defined by a @ 5-a6 splits. A separableextension over a non-commutative ring generalizes that over a commutative ring which wasdiscussed in [1]. Hirata introduced anOther kind of separable extensions called H-separabeones (see [2]). A/R is said to be H-separable if A @R A is isomorphic as an A-bimoduleto a direct sumrnand of A". riom {2, Theor…     

关于G－分次环与G－集的Smash积的几个结果

设Ｇ为任意群，本文借助于环的矩阵表示给出了Ｇ－分次环与任意可迁Ｇ－集的Ｓｍａｓｈ积是素环或单环的刻画。     

A Duality Theorem Associated with the Smash Product of Graded Ring and G-set

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