In this paper, we provide a diagrammatic approach to study the branching rules for cell modules on a sequence of walled Brauer algebras. This approach also allows us to calculate the structure constants of multiplicat...In this paper, we provide a diagrammatic approach to study the branching rules for cell modules on a sequence of walled Brauer algebras. This approach also allows us to calculate the structure constants of multiplication over the Grothendieck ring of the sequence.展开更多
We construct the Grothendieck rings of a class of 2n^(2)dimensional semisimple Hopf Algebras H_(2n)^(2),which can be viewed as a generalization of the 8 dimensional Kac-Paljutkin Hopf algebra H8.All irreducible H_(2n)...We construct the Grothendieck rings of a class of 2n^(2)dimensional semisimple Hopf Algebras H_(2n)^(2),which can be viewed as a generalization of the 8 dimensional Kac-Paljutkin Hopf algebra H8.All irreducible H_(2n)^(2)-modules are classified.Furthermore,we describe the Grothendieck rings r(H_(2n)^(2))by generators and relations explicitly.展开更多
We introduce the zero-divisor graph for an abelian regular ring and show that if R,S are abelian regular, then (K0(R),[R])≌(K0(S),[S]) if and only if they have isomorphic reduced zero-divisor graphs. It is shown that...We introduce the zero-divisor graph for an abelian regular ring and show that if R,S are abelian regular, then (K0(R),[R])≌(K0(S),[S]) if and only if they have isomorphic reduced zero-divisor graphs. It is shown that the maximal right quotient ring of a potent semiprimitive normal ring is abelian regular, moreover, the zero-divisor graph of such a ring is studied.展开更多
Let R be a ring with an identity element.R∈IBN means that R<sup>m</sup>■R<sup>n</sup> implies m=n,R ∈IBN<sub>1</sub> means that R<sup>m</sup> ■R<sup>n</sup&...Let R be a ring with an identity element.R∈IBN means that R<sup>m</sup>■R<sup>n</sup> implies m=n,R ∈IBN<sub>1</sub> means that R<sup>m</sup> ■R<sup>n</sup>⊕K implies m≥n,and R ∈IBN<sub>2</sub> means that R<sup>m</sup>■R<sup>m</sup>⊕K implies K=0.In this paper we give some characteristic properties of IBN<sub>1</sub> and IBN<sub>2</sub>,with orderings o the Grothendieck groups.In addition,we obtain the following results:（1）If R ∈IBM<sub>1</sub> and all finitely generated projective left R-modules are stably free,then the Grothendieck group K<sub>o</sub>（R）is a totally ordered abelian group.（2）If the pre-ordering of the Grothendieck group K<sub>o</sub>（R）of a ring R is a partial ordering,then R ∈IBM<sub>1</sub> or K<sub>o</sub>（R）=0.展开更多
In 1955, Serre gave an open problem on freeness of projective modules over polynomial ring over a field. In fact, it was proved by Serre in 1958 that if P is a finitely generated projective module over R[x<sub>1...In 1955, Serre gave an open problem on freeness of projective modules over polynomial ring over a field. In fact, it was proved by Serre in 1958 that if P is a finitely generated projective module over R[x<sub>1</sub>,…, x<sub>n</sub>], then P is stably isomorphic to a free module.展开更多
Let H = uq（sl（2）） or u（sl（2））. By means of the standard basis of polynomial algebras, the Glebsch-Gordan formula and quantum Clebsch-Gordan formula are proved by a unified method, and the explicit formula of t...Let H = uq（sl（2）） or u（sl（2））. By means of the standard basis of polynomial algebras, the Glebsch-Gordan formula and quantum Clebsch-Gordan formula are proved by a unified method, and the explicit formula of the decomposition of V（1）^n into the direct sum of simple modules is given in this paper.展开更多
文摘In this paper, we provide a diagrammatic approach to study the branching rules for cell modules on a sequence of walled Brauer algebras. This approach also allows us to calculate the structure constants of multiplication over the Grothendieck ring of the sequence.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.11671024,11701019,11871301)the Science and Technology Project of Beijing Municipal Education Commission(Grant No.KM202110005012).
文摘We construct the Grothendieck rings of a class of 2n^(2)dimensional semisimple Hopf Algebras H_(2n)^(2),which can be viewed as a generalization of the 8 dimensional Kac-Paljutkin Hopf algebra H8.All irreducible H_(2n)^(2)-modules are classified.Furthermore,we describe the Grothendieck rings r(H_(2n)^(2))by generators and relations explicitly.
基金Partially supported by the NSF (10071035) of China.
文摘We introduce the zero-divisor graph for an abelian regular ring and show that if R,S are abelian regular, then (K0(R),[R])≌(K0(S),[S]) if and only if they have isomorphic reduced zero-divisor graphs. It is shown that the maximal right quotient ring of a potent semiprimitive normal ring is abelian regular, moreover, the zero-divisor graph of such a ring is studied.
基金Supported by National Nature Science Foundation of China.
文摘Let R be a ring with an identity element.R∈IBN means that R<sup>m</sup>■R<sup>n</sup> implies m=n,R ∈IBN<sub>1</sub> means that R<sup>m</sup> ■R<sup>n</sup>⊕K implies m≥n,and R ∈IBN<sub>2</sub> means that R<sup>m</sup>■R<sup>m</sup>⊕K implies K=0.In this paper we give some characteristic properties of IBN<sub>1</sub> and IBN<sub>2</sub>,with orderings o the Grothendieck groups.In addition,we obtain the following results:（1）If R ∈IBM<sub>1</sub> and all finitely generated projective left R-modules are stably free,then the Grothendieck group K<sub>o</sub>（R）is a totally ordered abelian group.（2）If the pre-ordering of the Grothendieck group K<sub>o</sub>（R）of a ring R is a partial ordering,then R ∈IBM<sub>1</sub> or K<sub>o</sub>（R）=0.
基金Project supported by the National Natural Science Foundation of China
文摘In 1955, Serre gave an open problem on freeness of projective modules over polynomial ring over a field. In fact, it was proved by Serre in 1958 that if P is a finitely generated projective module over R[x<sub>1</sub>,…, x<sub>n</sub>], then P is stably isomorphic to a free module.
基金the National Natural Science Foundation of China （No. 10471121 10771182+2 种基金 10771183） Sino-German Project （No. GZ310） Nanjing Agricultural University Youth Science and Technology Innovation Foundation （No. KJ05028）.
文摘Let H = uq（sl（2）） or u（sl（2））. By means of the standard basis of polynomial algebras, the Glebsch-Gordan formula and quantum Clebsch-Gordan formula are proved by a unified method, and the explicit formula of the decomposition of V（1）^n into the direct sum of simple modules is given in this paper.
