高维的Frobenius型半单Hopf代数

设H是特征为零的代数闭域k上的半单Hopf代数.本文证明了如果dimкH 是小于351的奇数,则H是Frobenius型Hopf代数.     

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   

Cleft扩张下表示的不变性质

讨论半单Hopf代数上经cleft扩张后的代数表示的不变性质.首先,解释了cleft扩张的概念,并给出cleft扩张和交叉积之间的联系(交叉积是解决问题的基本方法).然后,利用这些关系证明了:当k是代数闭域,H是一个有限维的半单k代数时,对一个有限维k代数,其H-cleft扩张下的代数表示型是不变的.另一方面证明了:当k是任意域,H是一个有限维的半单k-Hopf代数,对一个根是H稳定的k代数,其Nakayama性质在H-cleft扩张下也是不变的.     

半单弱Hopf代数的作用

令H是半单弱Hopf代数, A是左H-模代数.我们证明了正则A-模的内射维数, A#H-模A的内射维数和正则A#H-模的内射维数三者是相等的. 而且,利用H在A上的不动点代数我们给出了A是Gorenstein代数的充要条件.     

AS-Gorenstein代数的Yoneda代数

令A为诺特基本k-代数,设J为其Jacobson根且半单代数A/J同构于有限个k的直积.证明了如果A是AS-Gorenstein代数,则其Yoneda代数Ext*A(A/J,A/J)是Frobenius代数;如果A的内射维数injdimAA=d,则函子ExtdA(-,A)是可表示的.     

3g3维半单Hopf代数的结构

设g为素数,k是特征为零的代数闭域,日是k上的3q3维半单Hopf代数．本文证明了日总是半可解的,即H可由群代数或对偶群代数经过扩张得到．     

Gorenstein代数上的Hopf代数作用

令H是有限维Hopf代数，A是左H-模代数。本文证明了A是Gorenstein代数的充分必要条件。A^H也是Gorenstein代数的条件。它是Enochs EE,GarciaJJ和del RioA关于群作用相应的理论的推广，同时给出A/A^H是Frobenius扩张的条件。     

Smash积的复杂度

给定任意一个有限维代数A,记其复杂度为C（A）.本文的主要结果是：如果有限维Hopf代数H和H是半单的,则对任意有限维H-模代数A,有C（A#H）=C（A）.利用此等式,可以计算一些代数的复杂度.     

单的模代数及其稳定化子

研究了有限维Hopf代数H与其单的模代数A的smash积的结构.通过给出A的反代数与其极小左理想的稳定化子的结构,证明了H与A的smash积与某个代数上的全矩阵代数是代数同构的,推广了以往的结果.     

Hopf代数的半单扩张与不变子环的同调维数

设H是域k上的有限维Hopf代数,A是左H-模代数,AH是A的H-不变子环.假定A/AH是半单扩张且A是平坦的右AH-模.如果H*是unimodular,且存在c∈C(A),使t·c=1.我们证明了WD(AH)=WD(A)=WD(A#H).此外,如果A是投射的左及右AH-模,则有LD(AH)=LD(A)=LD(A#H).     

低维的Frobenius 型半单Hopf 代数

Let k be an algebraically closed field of characteristic zero.This paper proves that semisimple Hopf algebras over k of dimension 66,70 and 78 are of Frobenius type.     

Commutator Hopf Subalgebras and Irreducible Representations

Montgomery and Witherspoon proved that upper and lower semisolvable, semisimple, finite dimensional Hopf algebras are of Frobenius type when their dimensions are not divisible by the characteristic of the base field. In this note we show that a finite dimensional, semisimple, lower solvable Hopf algebra is always of Frobenius type, in arbitrary characteristic.     

Relations Between Algebras and Their Subalgebras of Invariants

We prove that, if H is a finite-dimensional semisimple Hopf algebra, and A is an FCR H-module algebra over an algebraically closed field, then A is a PI-algebra, provided the subalgebra of invariants is a PI-algebra. We also show that if A is an affine algebra with an action of a finite group G by automorphisms, the subalgebra of the fixed points A^G is in the center of A, and the characteristic of the ground field is either zero or relatively prime to the order of G, then A^G is affine. Analogous results are proved for graded algebras and H-module algebras over a semisimple triangular Hopf algebra over a field of characteristic zero. We prove also that, if A is an H-module algebra with an identity element, and H is either a semisimple group algebra or its dual, then, if A is semiprimitive (semiprime), then so is A^H.     

Representation dimension for Hopf actions

Let H be a finite-dimensional Hopf algebra and assume that both H and H* are semisimple.The main result of this paper is to show that the representation dimension is an invariant under cleft extensions of H,that is,rep.dim(A) = rep.dim(A# σ H).Some of the applications of this equality are also given.     

On Semisimple Hopf Algebras of Dimension pq r

Let p and q be distinct prime numbers. We prove a result on the existence of nontrivial group-like elements in a certain class of semisimple Hopf algebras of dimension pq ^r . We conclude the classification of semisimple Hopf algebras A of dimension pq ² over an algebraically closed field k of characteristic zero, such that both A and A ^* are of Frobenius type. We also complete the classification of semisimple Hopf algebras of dimension pq ²<100.     

Invariant properties of representations under cleft extensions

The main aim of this paper is to give the invariant properties of representations of algebras under cleft extensions over a semisimple Hopf algebra. Firstly, we explain the concept of the cleft extension and give a relation between the cleft extension and the crossed product which is the approach we depend upon. Then, by making use of them, we prove that over an algebraically closed field k, for a finite dimensional Hopf algebra H which is semisimple as well as its dual H*, the representation type of an algebra is an invariant property under a finite dimensional H-cleft extension . In the other part, we still show that over an arbitrary field k, the Nakayama property of a k-algebra is also an invariant property under an H -cleft extension when the radical of the algebra is H-stable.