Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1,…, un) of rational functions of n independent indeterminates u1,…,un.It is an i...Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1,…, un) of rational functions of n independent indeterminates u1,…,un.It is an isomorphism between two cluster algebras associated to the matrix A (see sec. 4 for the precise meaning). When A is of finite type, these isomorphisms behave nicely; they are compatible with the BGP-reflection functors of cluster categories defined in a previous work if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras, and they are also compatible with the 'truncated simple reflections' defined by Fomin-Zelevinsky. Using the construction of preprojective or preinjective modules of hereditary algebras by DIab-Ringel and the Coxeter automorphisms (i.e. a product of these isomorphisms), we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types.展开更多
Let U be a quantized enveloping algebra and U its modified form. Lusztig gives some symmetries on U and U. In view of the realization of U by the reduced Drinfeld double of the Ringel- Hall algebra, one can apply the ...Let U be a quantized enveloping algebra and U its modified form. Lusztig gives some symmetries on U and U. In view of the realization of U by the reduced Drinfeld double of the Ringel- Hall algebra, one can apply the BGP-refiection functors to the double Ringel-HM1 algebra to obtain Lusztig＇s symmetries on U and their important properties, for instance, the braid relations. In this paper, we define a modified form Hof the Ringel-Hall algebra and realize the Lusztig＇s symmetries on U by applying the BGP-reflection functors to H展开更多
We define the BGP-reflection functors in the derived categories and the root categories. By Ringel's Hall algebra approach, the BGP-reflection functor is applicable to obtain the classical Weyl group action on the...We define the BGP-reflection functors in the derived categories and the root categories. By Ringel's Hall algebra approach, the BGP-reflection functor is applicable to obtain the classical Weyl group action on the Lie algebra.展开更多
基金This work was supported by the National Natural Science Foundation of China(Grant No.10471071)partially by the Cultivation Fund of the Key Scientific and Technical Innovation Project,Ministry of Education of China 2005.
文摘Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1,…, un) of rational functions of n independent indeterminates u1,…,un.It is an isomorphism between two cluster algebras associated to the matrix A (see sec. 4 for the precise meaning). When A is of finite type, these isomorphisms behave nicely; they are compatible with the BGP-reflection functors of cluster categories defined in a previous work if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras, and they are also compatible with the 'truncated simple reflections' defined by Fomin-Zelevinsky. Using the construction of preprojective or preinjective modules of hereditary algebras by DIab-Ringel and the Coxeter automorphisms (i.e. a product of these isomorphisms), we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types.
基金Supported by National Natural Science Foundation of China(Grant No.11131001)the Fundamental Research Funds for the Central Universities(Grant No.BLX2013014)
文摘Let U be a quantized enveloping algebra and U its modified form. Lusztig gives some symmetries on U and U. In view of the realization of U by the reduced Drinfeld double of the Ringel- Hall algebra, one can apply the BGP-refiection functors to the double Ringel-HM1 algebra to obtain Lusztig＇s symmetries on U and their important properties, for instance, the braid relations. In this paper, we define a modified form Hof the Ringel-Hall algebra and realize the Lusztig＇s symmetries on U by applying the BGP-reflection functors to H
基金supported by Doctoral Program Foundation of the Ministry of Education of China(2003)the 973 Project of the Ministry of Science and Technology of China(Grant No.1999075101).
文摘We define the BGP-reflection functors in the derived categories and the root categories. By Ringel's Hall algebra approach, the BGP-reflection functor is applicable to obtain the classical Weyl group action on the Lie algebra.
