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 re...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.展开更多
Let k be an algebraically closed field. It has been proved by Zhang and Xu that if a bocs is of tame representation type, then the degree of the differential of the first solid arrow must be less than or equal to 3. W...Let k be an algebraically closed field. It has been proved by Zhang and Xu that if a bocs is of tame representation type, then the degree of the differential of the first solid arrow must be less than or equal to 3. We will prove in the present paper that: The bocs is still wild when the degree of the differential of the first arrow is equal to 3. Especially, the bocs with only one solid arrow is of tame type if and only if the degree of the differential of the arrow is less than or equal to 2. Moreover, we classify in this case the growth problems of the representation category of the bocs and layout the sufficient and necessary conditions when the bocs is of finite representation type, tame domestic and tame exponential growth respectively.展开更多
The well-known tame theorem tells that for a given tame bocs and a positive integer n there exist finitely many minimal bocses, such that any representation of the original bocs of dimension at most n is isomorphic to...The well-known tame theorem tells that for a given tame bocs and a positive integer n there exist finitely many minimal bocses, such that any representation of the original bocs of dimension at most n is isomorphic to the image of a representation of some minimal bocses under a certain reduction functor. In the present paper we will give an alternative statement of the tame theorem in terms of matrix problem, by constructing a unified minimal matrix problem whose indecomposable matrices cover all the canonical forms of the indecomposable representations of dimension at most n for each non-negative integer n.展开更多
基金This work was partially supported by the Program for New Century Excellent Talents in University (Grant No.04-0522) the National Natural Science Foundation of China (Grant No.10571153).
文摘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.
基金supported by National Natural Science Foundation of China (Grant No. 10731070)
文摘Let k be an algebraically closed field. It has been proved by Zhang and Xu that if a bocs is of tame representation type, then the degree of the differential of the first solid arrow must be less than or equal to 3. We will prove in the present paper that: The bocs is still wild when the degree of the differential of the first arrow is equal to 3. Especially, the bocs with only one solid arrow is of tame type if and only if the degree of the differential of the arrow is less than or equal to 2. Moreover, we classify in this case the growth problems of the representation category of the bocs and layout the sufficient and necessary conditions when the bocs is of finite representation type, tame domestic and tame exponential growth respectively.
基金supported by National Natural Science Foundation of China (Grant Nos.10731070,10501010)
文摘The well-known tame theorem tells that for a given tame bocs and a positive integer n there exist finitely many minimal bocses, such that any representation of the original bocs of dimension at most n is isomorphic to the image of a representation of some minimal bocses under a certain reduction functor. In the present paper we will give an alternative statement of the tame theorem in terms of matrix problem, by constructing a unified minimal matrix problem whose indecomposable matrices cover all the canonical forms of the indecomposable representations of dimension at most n for each non-negative integer n.
