We consider the Borcherds-Cartan matrix obtained from a symmetrizable generalized Cartan matrix by adding one imaginary simple root. We extend the result of Gebert and Teschner [Lett. Math. Phys., 1994, 31： 327-334] ...We consider the Borcherds-Cartan matrix obtained from a symmetrizable generalized Cartan matrix by adding one imaginary simple root. We extend the result of Gebert and Teschner [Lett. Math. Phys., 1994, 31： 327-334] to the quantum case. Moreover, we give a connection between the irreducible dominant representations of quantum Kac-Moody algebras and those of quantum generalized Kac-Moody algebras. As the result, a large class of irreducible dominant representations of quantum generalized Kac-Moody algebras were obtained from representations of quantum Kac-Moody algebras through tensor algebras.展开更多
In this article,we give an explicit way to construct representations of thefundamental groupπ_(1)(X),where X is a hyperbolic curve over C.Our motivation isto study a special space in MDR(X,SL_(2)(C))which is called t...In this article,we give an explicit way to construct representations of thefundamental groupπ_(1)(X),where X is a hyperbolic curve over C.Our motivation isto study a special space in MDR(X,SL_(2)(C))which is called the space of permissibleconnections in Faltings(Compos Math 48(2):223-269,1983),or indigenous bundlesin Gunning(Math Ann 170:67-86,1967).We get representations by constructingHiggs bundles,and we show that the family we get intersects the space of permissibleconnections PC in a positive dimension.In this way,we actually get a deformation ofthe canonical representation in PC,and all these deformations are given by explicitconstructed Higgs bundles.We also estimate the dimension of this deformation space.展开更多
The notion of Higgs-de Rham flows was introduced by Lan et al.(2019),as an analogue of Yang-Mills-Higgs flows in the complex nonabelian Hodge theory.In this paper we investigate a small part of this theory,and study t...The notion of Higgs-de Rham flows was introduced by Lan et al.(2019),as an analogue of Yang-Mills-Higgs flows in the complex nonabelian Hodge theory.In this paper we investigate a small part of this theory,and study those Higgs-de Rham flows which are of level zero.We improve the original definition of level-zero Higgs-de Rham flows(which works for general levels),and establish a Hitchin-Simpson type correspondence between such objects and certain representations of fundamental groups in positive characteristic,which generalizes a classical results of Katz(1973).We compare the deformation theories of two sides in the correspondence,and translate the Galois action on the geometric fundamental groups of algebraic varieties defined over finite fields into the Higgs side.展开更多
基金The authors would like suggestions and express their sincere gratitude valuable discussion. The second author （Zhao） Natural Science Foundation of China （Grant No. Funds for the Central Universities. to thank the referees for many helpful to Professor Bangming Deng for many was partially supported by the National 11226063） and the Fundamental Research
文摘We consider the Borcherds-Cartan matrix obtained from a symmetrizable generalized Cartan matrix by adding one imaginary simple root. We extend the result of Gebert and Teschner [Lett. Math. Phys., 1994, 31： 327-334] to the quantum case. Moreover, we give a connection between the irreducible dominant representations of quantum Kac-Moody algebras and those of quantum generalized Kac-Moody algebras. As the result, a large class of irreducible dominant representations of quantum generalized Kac-Moody algebras were obtained from representations of quantum Kac-Moody algebras through tensor algebras.
文摘In this article,we give an explicit way to construct representations of thefundamental groupπ_(1)(X),where X is a hyperbolic curve over C.Our motivation isto study a special space in MDR(X,SL_(2)(C))which is called the space of permissibleconnections in Faltings(Compos Math 48(2):223-269,1983),or indigenous bundlesin Gunning(Math Ann 170:67-86,1967).We get representations by constructingHiggs bundles,and we show that the family we get intersects the space of permissibleconnections PC in a positive dimension.In this way,we actually get a deformation ofthe canonical representation in PC,and all these deformations are given by explicitconstructed Higgs bundles.We also estimate the dimension of this deformation space.
基金supported by National Natural Science Foundation of China(Grant Nos.11622109 and 11721101)Anhui Initiative in Quantum Information Technologies(Grant No.AHY150200)supported by One-Thousand-Talents Program of China。
文摘The notion of Higgs-de Rham flows was introduced by Lan et al.(2019),as an analogue of Yang-Mills-Higgs flows in the complex nonabelian Hodge theory.In this paper we investigate a small part of this theory,and study those Higgs-de Rham flows which are of level zero.We improve the original definition of level-zero Higgs-de Rham flows(which works for general levels),and establish a Hitchin-Simpson type correspondence between such objects and certain representations of fundamental groups in positive characteristic,which generalizes a classical results of Katz(1973).We compare the deformation theories of two sides in the correspondence,and translate the Galois action on the geometric fundamental groups of algebraic varieties defined over finite fields into the Higgs side.
