Let g be a simple complex Lie algebra of classical type with a Cartan subalgebraη.We fix a standard parabolic subalgebra p?η.The socular simple modules are just those highest weight modules with the largest possible...Let g be a simple complex Lie algebra of classical type with a Cartan subalgebraη.We fix a standard parabolic subalgebra p?η.The socular simple modules are just those highest weight modules with the largest possible Gelfand-Kirillov dimension in the corresponding parabolic category Op.In this article,we give an explicit characterization of these modules.When the module is integral,our characterization is given by the information of the corresponding Young tableau associated with the given highest weight module.When the module is nonintegral,we still have some characterization by using the results in the integral case.In our characterization,we define a particular Young diagram called the Z-diagram.From this diagram,we can describe the partition type of the unique Richardson orbit associated with the given parabolic subalgebra p.展开更多
During the last decade, a great deal of activity has been devoted to the calculation of the HilbertPoincar′e series of unitary highest weight representations and related modules in algebraic geometry. However,uniform...During the last decade, a great deal of activity has been devoted to the calculation of the HilbertPoincar′e series of unitary highest weight representations and related modules in algebraic geometry. However,uniform formulas remain elusive—even for more basic invariants such as the Gelfand-Kirillov dimension or the Bernstein degree, and are usually limited to families of representations in a dual pair setting. We use earlier work by Joseph to provide an elementary and intrinsic proof of a uniform formula for the Gelfand-Kirillov dimension of an arbitrary unitary highest weight module in terms of its highest weight. The formula generalizes a result of Enright and Willenbring(in the dual pair setting) and is inspired by Wang's formula for the dimension of a minimal nilpotent orbit.展开更多
In this paper,we construct a superfermionic representation as well as a vertex representation for twisted general linear affine Lie superalgebras.We also establish a module isomorphism between them,which generalizes t...In this paper,we construct a superfermionic representation as well as a vertex representation for twisted general linear affine Lie superalgebras.We also establish a module isomorphism between them,which generalizes the super boson-fermion correspondence of type B given by Kac-van de Leur.Based on this isomorphism,we determine explicitly the irreducible components of these two representations.Particularly,we obtain in this way two kinds of systematic construction of level 1 irreducible integrable highest weight modules for twisted general linear affine Lie superalgebras.展开更多
The simple modules for electrical Lie algebra of type D5 were investigated.The sufficient and necessary criteria of the simple Z-graded highest weight modules were established by means of determining the singular vect...The simple modules for electrical Lie algebra of type D5 were investigated.The sufficient and necessary criteria of the simple Z-graded highest weight modules were established by means of determining the singular vectors of the Verma modules.The simple highest weight module is isomorphic to either that for the symplectic Lie algebra sp4 or Verma module.展开更多
In this paper, the Harish-Chandra modules and Verma modules over Block algebra $ \mathfrak{L} $ [G] are investigated. More precisely, the irreducibility of the Verma modules over $ \mathfrak{L} $ [G] is completely det...In this paper, the Harish-Chandra modules and Verma modules over Block algebra $ \mathfrak{L} $ [G] are investigated. More precisely, the irreducibility of the Verma modules over $ \mathfrak{L} $ [G] is completely determined, and the Harish-Chandra modules over $ \mathfrak{L} $ [?] are classified.展开更多
It is known that the Schrddinger-Virasoro algebras, including the original Schrddinger-Virasoro algebra and the twisted Schr?dinger-Virasoro algebra, are playing important roles in mathematics and statistical physics....It is known that the Schrddinger-Virasoro algebras, including the original Schrddinger-Virasoro algebra and the twisted Schr?dinger-Virasoro algebra, are playing important roles in mathematics and statistical physics. In this paper, we study the tensor products of weight modules over the Schr?dinger- Virasoro algebras. The irreducibility criterion for the tensor products of highest weight modules with intermediate series modules over the Schr?dinger-Virasoro algebra is obtained.展开更多
In this paper,we consider the imaginary highest weight modules and the imaginary Whittaker modules for the affine Nappi-Witten algebra.We show that simple singular imaginary Whittaker modules at level(κ、c)(κ∈C~*)a...In this paper,we consider the imaginary highest weight modules and the imaginary Whittaker modules for the affine Nappi-Witten algebra.We show that simple singular imaginary Whittaker modules at level(κ、c)(κ∈C~*)are simple imaginary highest weight modules.The necessary and sufficient conditions for these imaginary modules to be simple are given.All simple imaginary modules are classified.展开更多
First, the authors give a GrSbner-Shirshov basis of the finite-dimensional irre- ducible module Vq（λ） of the Drinfeld-Jimbo quantum group Uq（G2） by using the double free module method and the known GrSbner-Shirsh...First, the authors give a GrSbner-Shirshov basis of the finite-dimensional irre- ducible module Vq（λ） of the Drinfeld-Jimbo quantum group Uq（G2） by using the double free module method and the known GrSbner-Shirshov basis of Uq（G2）. Then, by specializing a suitable version of Uq （G2） at q = 1, they get a GrSbner-Shirshov basis of the universal enveloping algebra U（G2） of the simple Lie algebra of type G2 and the finite-dimensional irreducible U（G2）-module V（λ）.展开更多
There are no simple singular Whittaker modules over most of important algebras,such as simple complex finite-dimensional Lie algebras,affine Kac-Moody Lie algebras,the Virasoro algebra,the Heisenberg-Virasoro algebra ...There are no simple singular Whittaker modules over most of important algebras,such as simple complex finite-dimensional Lie algebras,affine Kac-Moody Lie algebras,the Virasoro algebra,the Heisenberg-Virasoro algebra and the Schrödinger-Witt algebra.In this paper,however,we construct simple singular Whittaker modules over the Schrödinger algebra.Moreover,simple singular Whittaker modules over the Schrödinger algebra are classified.As a result,simple modules for the Schrödinger algebrawhich are locally finite over the positive part are completely classified.We also give characterizations of simple highestweight modules and simple singular Whittaker modules.展开更多
Let y be a generalized Kac-Moody algebra with an integral Borcherds-Cartan matrix. In this paper, we define a d-type weak quantum generalized Kac-Moody algebra wUq^d（y）, which is a weak Hopf algebra. We also study t...Let y be a generalized Kac-Moody algebra with an integral Borcherds-Cartan matrix. In this paper, we define a d-type weak quantum generalized Kac-Moody algebra wUq^d（y）, which is a weak Hopf algebra. We also study the highest weight module over the weak quantum algebra wUdq^d（y） and weak A-forms of wUq^d（y）.展开更多
In this paper,we prove that the electrical Lie algebra e D_(5)is isomorphic to the semidirect product of sp_(4)and a 2-step nilpotent Lie algebra.Furthermore,we classify the irreducible highest weight modules for e D_...In this paper,we prove that the electrical Lie algebra e D_(5)is isomorphic to the semidirect product of sp_(4)and a 2-step nilpotent Lie algebra.Furthermore,we classify the irreducible highest weight modules for e D_(5).展开更多
In this paper, an explicit determinant formula is given for the Verma modules over the Lie algebra W(2, 2). We construct a natural realization of a certain vaccum module for the algebra W(2, 2) via the Weyl vertex alg...In this paper, an explicit determinant formula is given for the Verma modules over the Lie algebra W(2, 2). We construct a natural realization of a certain vaccum module for the algebra W(2, 2) via the Weyl vertex algebra. We also describe several results including the irreducibility, characters and the descending filtrations of submodules for the Verma module over the algebra W(2, 2).展开更多
基金supported by National Natural Science Foundation of China(Grant No.12171344)the National Key R&D Program of China(Grant Nos.2018YFA0701700 and 2018YFA0701701)。
文摘Let g be a simple complex Lie algebra of classical type with a Cartan subalgebraη.We fix a standard parabolic subalgebra p?η.The socular simple modules are just those highest weight modules with the largest possible Gelfand-Kirillov dimension in the corresponding parabolic category Op.In this article,we give an explicit characterization of these modules.When the module is integral,our characterization is given by the information of the corresponding Young tableau associated with the given highest weight module.When the module is nonintegral,we still have some characterization by using the results in the integral case.In our characterization,we define a particular Young diagram called the Z-diagram.From this diagram,we can describe the partition type of the unique Richardson orbit associated with the given parabolic subalgebra p.
基金supported by National Natural Science Foundation of China(Grant No.11171324)the Hong Kong Research Grants Council under RGC Project(Grant No.60311)the Hong Kong University of Science and Technology under DAG S09/10.SC02.
文摘During the last decade, a great deal of activity has been devoted to the calculation of the HilbertPoincar′e series of unitary highest weight representations and related modules in algebraic geometry. However,uniform formulas remain elusive—even for more basic invariants such as the Gelfand-Kirillov dimension or the Bernstein degree, and are usually limited to families of representations in a dual pair setting. We use earlier work by Joseph to provide an elementary and intrinsic proof of a uniform formula for the Gelfand-Kirillov dimension of an arbitrary unitary highest weight module in terms of its highest weight. The formula generalizes a result of Enright and Willenbring(in the dual pair setting) and is inspired by Wang's formula for the dimension of a minimal nilpotent orbit.
基金supported by National Natural Science Foundation of China(Grant No.12161141001)the Fundamental Research Funds for the Central Universities(Grant No.20720230020)S.Tan is supported by National Natural Science Foundation of China(Grant No.12131018)。
文摘In this paper,we construct a superfermionic representation as well as a vertex representation for twisted general linear affine Lie superalgebras.We also establish a module isomorphism between them,which generalizes the super boson-fermion correspondence of type B given by Kac-van de Leur.Based on this isomorphism,we determine explicitly the irreducible components of these two representations.Particularly,we obtain in this way two kinds of systematic construction of level 1 irreducible integrable highest weight modules for twisted general linear affine Lie superalgebras.
基金Fundamental Research Funds for the Central Universities,China(No.2232021G13)。
文摘The simple modules for electrical Lie algebra of type D5 were investigated.The sufficient and necessary criteria of the simple Z-graded highest weight modules were established by means of determining the singular vectors of the Verma modules.The simple highest weight module is isomorphic to either that for the symplectic Lie algebra sp4 or Verma module.
基金supported by the Research Foundation for Postdoctor Programmethe National NaturalScience Foundation of China (Grant No. 10601057)
文摘In this paper, the Harish-Chandra modules and Verma modules over Block algebra $ \mathfrak{L} $ [G] are investigated. More precisely, the irreducibility of the Verma modules over $ \mathfrak{L} $ [G] is completely determined, and the Harish-Chandra modules over $ \mathfrak{L} $ [?] are classified.
基金the National Natural Science Foundation of China(Grant Nos.11571145,11871249)the Natural Science Foundation of Zhejiang Province(No.LZ14A010001).
文摘It is known that the Schrddinger-Virasoro algebras, including the original Schrddinger-Virasoro algebra and the twisted Schr?dinger-Virasoro algebra, are playing important roles in mathematics and statistical physics. In this paper, we study the tensor products of weight modules over the Schr?dinger- Virasoro algebras. The irreducibility criterion for the tensor products of highest weight modules with intermediate series modules over the Schr?dinger-Virasoro algebra is obtained.
基金Supported by NSF of China(Grant Nos.11801117,11801390)the Natural Science Foundation of Guangdong Province,China(Grant No.2018A030313268)the General Finacial Grant from the China Postdoctoral Science Foundation(Grant No.2016M600140)。
文摘In this paper,we consider the imaginary highest weight modules and the imaginary Whittaker modules for the affine Nappi-Witten algebra.We show that simple singular imaginary Whittaker modules at level(κ、c)(κ∈C~*)are simple imaginary highest weight modules.The necessary and sufficient conditions for these imaginary modules to be simple are given.All simple imaginary modules are classified.
基金supported by the National Natural Science Foundation of China(Nos.11061033,11361056)
文摘First, the authors give a GrSbner-Shirshov basis of the finite-dimensional irre- ducible module Vq（λ） of the Drinfeld-Jimbo quantum group Uq（G2） by using the double free module method and the known GrSbner-Shirshov basis of Uq（G2）. Then, by specializing a suitable version of Uq （G2） at q = 1, they get a GrSbner-Shirshov basis of the universal enveloping algebra U（G2） of the simple Lie algebra of type G2 and the finite-dimensional irreducible U（G2）-module V（λ）.
文摘There are no simple singular Whittaker modules over most of important algebras,such as simple complex finite-dimensional Lie algebras,affine Kac-Moody Lie algebras,the Virasoro algebra,the Heisenberg-Virasoro algebra and the Schrödinger-Witt algebra.In this paper,however,we construct simple singular Whittaker modules over the Schrödinger algebra.Moreover,simple singular Whittaker modules over the Schrödinger algebra are classified.As a result,simple modules for the Schrödinger algebrawhich are locally finite over the positive part are completely classified.We also give characterizations of simple highestweight modules and simple singular Whittaker modules.
基金Supported in part by the Scientific Research Foundation of Zhejiang Provincial Education Department under grant number 20040322It is also sponsored by SRF for ROCS,SEM
文摘Let y be a generalized Kac-Moody algebra with an integral Borcherds-Cartan matrix. In this paper, we define a d-type weak quantum generalized Kac-Moody algebra wUq^d（y）, which is a weak Hopf algebra. We also study the highest weight module over the weak quantum algebra wUdq^d（y） and weak A-forms of wUq^d（y）.
基金Supported by the Fundamental Research Funds for the Central Universities(Grant No.2232021G-13).
文摘In this paper,we prove that the electrical Lie algebra e D_(5)is isomorphic to the semidirect product of sp_(4)and a 2-step nilpotent Lie algebra.Furthermore,we classify the irreducible highest weight modules for e D_(5).
基金supported by National Natural Science Foundation of China (Grant Nos. 11271056, 11671056 and 11101030)National Science Foundation of Jiangsu (Grant No. BK20160403)+3 种基金National Science Foundation of Zhejiang (Grant Nos. LQ12A01005 and LZ14A010001)National Science Foundation of Shanghai (Grant No. 16ZR1425000)Beijing Higher Education Young Elite Teacher ProjectMorningside Center of Mathematics
文摘In this paper, an explicit determinant formula is given for the Verma modules over the Lie algebra W(2, 2). We construct a natural realization of a certain vaccum module for the algebra W(2, 2) via the Weyl vertex algebra. We also describe several results including the irreducibility, characters and the descending filtrations of submodules for the Verma module over the algebra W(2, 2).
