Let G be a locally compact unimodular group with Haar measure rmdx and ω be the Beurling's weight function on G (Reiter, [10]). In this paper the authors define a space Aωp,q (G) and prove that Aωp,q (G) is a t...Let G be a locally compact unimodular group with Haar measure rmdx and ω be the Beurling's weight function on G (Reiter, [10]). In this paper the authors define a space Aωp,q (G) and prove that Aωp,q (G) is a translation invariant Banach space. Fur- thermore the authors discuss inclusion properties and show that if G is a locally compact abelian group then Aωp,q (G) admits an approximate identity bounded in Lω1 (G). It is also proved that the space Lωp (G) Lω1 Lωq (G) is isometrically isomorphic to the space Aωp,q (G) and the space of multipliers from Lωp (G) to Lq-1, (G) is isometrically isomorphic to the dual of the space Aωp,q (G) iff G satisfies a property Ppq. At the end of this work it is showed that if G is a locally compact abelian group then the space of all multipliers from Lω1 (G) to Aωp,q (G) is the space Aωp,q (G).展开更多
In this paper,we classify the simple uniformly bounded weight modules for the vector eld Lie algebra W1 of in nite rank.It turns out that any such modules are intermediate series modules.This result is very di erent f...In this paper,we classify the simple uniformly bounded weight modules for the vector eld Lie algebra W1 of in nite rank.It turns out that any such modules are intermediate series modules.This result is very di erent from the vector eld Lie algebra Wd of nite rank.展开更多
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
Resistive random-access memory(RRAM)is a promising technology to develop nonvolatile memory and artificial synaptic devices for brain-inspired neuromorphic computing.Here,we have developed a STO:Ag/SiO_(2) bilayer bas...Resistive random-access memory(RRAM)is a promising technology to develop nonvolatile memory and artificial synaptic devices for brain-inspired neuromorphic computing.Here,we have developed a STO:Ag/SiO_(2) bilayer based memristor that has exhibited a filamentary resistive switching with stable endurance and long-term data retention ability.The memristor also exhibits a tunable resistance modulation under positive and negative pulse trains,which could fully mimic the potentiation and depression behavior like a bio-synapse.Several synaptic plasticity functions,including long-term potentiation(LTP)and long-term depression(LTD),paired-pulsed facilitation(PPF),spike-rate-dependent-plasticity(SRDP),and post-tetanic potentiation(PTP),are faithfully implemented with the fabricated memristor.Moreover,to demonstrate the feasibility of our memristor synapse for neuromorphic applications,spike-timedependent plasticity(STDP)is also investigated.Based on conductive atomic force microscopy observations and electrical transport model analyses,it can be concluded that it is the controlled formation and rupture of Ag filaments that are responsible for the resistive switching while exhibiting a switching ratio of~10;along with a good endurance and stability suitable for nonvolatile memory applications.Before fully electroforming,the gradual conductance modulation of Ag/STO:Ag/SiO_(2)/p^(++)-Si memristor can be realized,and the working mechanism could be explained by the succeeding growth and contraction of Ag filaments promoted by a redox reaction.This newly fabricated memristor may enable the development of nonvolatile memory and realize controllable resistance/weight modulation when applied as an artificial synapse for neuromorphic computing.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,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
We show that the support of an irreducible weight module over the twisted Heisenberg-Virasoro algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all nontrivial weight s...We show that the support of an irreducible weight module over the twisted Heisenberg-Virasoro algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain that every irreducible weight module over the twisted Heisenber-Virasoro algebra, having a nontrivial finite-dimensional weight space, is a Harish-Chandra module （and hence is either an irreducible highest or lowest weight module or an irreducible module from the intermediate series）.展开更多
It is shown that the support of an irreducible weight module over the SchrSdinger-Virasoro Lie algebra with an infinite-dimensional weight space coincides with the weight lattice, and all nontrivial weight spaces of s...It is shown that the support of an irreducible weight module over the SchrSdinger-Virasoro Lie algebra with an infinite-dimensional weight space coincides with the weight lattice, and all nontrivial weight spaces of such a module are infinite-dimensional. As a by-product, it is obtained that every simple weight module over Lie algebra of this type with a nontrivial finite-dimensional weight space is a Harish-Chandra module.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, 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
There are two extensions of Virasoro algebra with particular importance in superstring theory： the Ramond algebra and the Neveu-Schwarz algebra, which are Z2-graded extensions of the Virasoro algebra. In this paper, ...There are two extensions of Virasoro algebra with particular importance in superstring theory： the Ramond algebra and the Neveu-Schwarz algebra, which are Z2-graded extensions of the Virasoro algebra. In this paper, we show that the support of a simple weight module over the Ramond algebra with an infinite-dimensional weight space coincides with the weight lattice and that all intersections of non-trivial weight spaces and odd part or even part of the module are infinite-dimensional. This result together with the one that we have obtained over the Neveu-Schwarz algebra generalizes the result for the Virasoro algebra to the super-Virasoro algebras.展开更多
It is shown that there are no simple mixed modules over the twisted N = 1 Schrodinger-Neveu-Schwarz algebra, which implies that every irreducible weight module over it with a non-trivial finite-dimensional weight spac...It is shown that there are no simple mixed modules over the twisted N = 1 Schrodinger-Neveu-Schwarz algebra, which implies that every irreducible weight module over it with a non-trivial finite-dimensional weight space is a Harish-Chandra module.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
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
For any module V over the two-dimensional non-abelian Lie algebra b and scalar a ∈C, we define a class of weight modules Fα（V） with zero central charge over the affine Lie algebra A（1）. These weight modules have...For any module V over the two-dimensional non-abelian Lie algebra b and scalar a ∈C, we define a class of weight modules Fα（V） with zero central charge over the affine Lie algebra A（1）. These weight modules have infinite- dimensional weight spaces if and only if V is infinite dimensional. In this paper, we will determine necessary and sufficient conditions for these modules Fα（V） to be irreducible. In this way, we obtain a lot of irreducible weight A1（1）-modules with infinite-dimensional weight spaces.展开更多
The Heisenberg double D_(q)(E_(2))of the quantum Euclidean group O_(q)(E_(2))is the smash product of O_(q)(E_(2))with its Hopf dual U_(q)(e_(2)).For the algebra D_(q)(E_(2)),explicit descriptions of its prime,primitiv...The Heisenberg double D_(q)(E_(2))of the quantum Euclidean group O_(q)(E_(2))is the smash product of O_(q)(E_(2))with its Hopf dual U_(q)(e_(2)).For the algebra D_(q)(E_(2)),explicit descriptions of its prime,primitive and maximal spectra are obtained.All the prime factors of D_(q)(E_(2))are presented as generalized Weyl algebras.As a result,we obtain that the algebra D_(q)(E_(2))has no finite-dimensional representations,and D_(q)(E_(2))cannot have a Hopf algebra structure.The automorphism groups of the quantum Euclidean group and its Heisenberg double are determined.Some centralizers are explicitly described via generators and defining relations.This enables us to give a classification of simple weight modules and the so-called a-weight modules over the algebra D_(q)(E_(2)).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
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
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
文摘Let G be a locally compact unimodular group with Haar measure rmdx and ω be the Beurling's weight function on G (Reiter, [10]). In this paper the authors define a space Aωp,q (G) and prove that Aωp,q (G) is a translation invariant Banach space. Fur- thermore the authors discuss inclusion properties and show that if G is a locally compact abelian group then Aωp,q (G) admits an approximate identity bounded in Lω1 (G). It is also proved that the space Lωp (G) Lω1 Lωq (G) is isometrically isomorphic to the space Aωp,q (G) and the space of multipliers from Lωp (G) to Lq-1, (G) is isometrically isomorphic to the dual of the space Aωp,q (G) iff G satisfies a property Ppq. At the end of this work it is showed that if G is a locally compact abelian group then the space of all multipliers from Lω1 (G) to Aωp,q (G) is the space Aωp,q (G).
文摘In this paper,we classify the simple uniformly bounded weight modules for the vector eld Lie algebra W1 of in nite rank.It turns out that any such modules are intermediate series modules.This result is very di erent from the vector eld Lie algebra Wd of nite rank.
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financially supported by the National Science Funds for Excellent Young Scholars of China(no.61822106)the Natural Science Foundation of China(no.U19A2070)。
文摘Resistive random-access memory(RRAM)is a promising technology to develop nonvolatile memory and artificial synaptic devices for brain-inspired neuromorphic computing.Here,we have developed a STO:Ag/SiO_(2) bilayer based memristor that has exhibited a filamentary resistive switching with stable endurance and long-term data retention ability.The memristor also exhibits a tunable resistance modulation under positive and negative pulse trains,which could fully mimic the potentiation and depression behavior like a bio-synapse.Several synaptic plasticity functions,including long-term potentiation(LTP)and long-term depression(LTD),paired-pulsed facilitation(PPF),spike-rate-dependent-plasticity(SRDP),and post-tetanic potentiation(PTP),are faithfully implemented with the fabricated memristor.Moreover,to demonstrate the feasibility of our memristor synapse for neuromorphic applications,spike-timedependent plasticity(STDP)is also investigated.Based on conductive atomic force microscopy observations and electrical transport model analyses,it can be concluded that it is the controlled formation and rupture of Ag filaments that are responsible for the resistive switching while exhibiting a switching ratio of~10;along with a good endurance and stability suitable for nonvolatile memory applications.Before fully electroforming,the gradual conductance modulation of Ag/STO:Ag/SiO_(2)/p^(++)-Si memristor can be realized,and the working mechanism could be explained by the succeeding growth and contraction of Ag filaments promoted by a redox reaction.This newly fabricated memristor may enable the development of nonvolatile memory and realize controllable resistance/weight modulation when applied as an artificial synapse for neuromorphic computing.
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 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NSF Grants 10471096,10571120 of China"One Hundred Talents Program"from the University of Science and Technology of China
文摘We show that the support of an irreducible weight module over the twisted Heisenberg-Virasoro algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain that every irreducible weight module over the twisted Heisenber-Virasoro algebra, having a nontrivial finite-dimensional weight space, is a Harish-Chandra module （and hence is either an irreducible highest or lowest weight module or an irreducible module from the intermediate series）.
基金Supported by China Postdoctoral Science Foundation Grant 20080440720, NSF Grants 10671027, 10825101 of China and ＂One Hundred Talents Program＂ from University of Science and Technology of China
文摘It is shown that the support of an irreducible weight module over the SchrSdinger-Virasoro Lie algebra with an infinite-dimensional weight space coincides with the weight lattice, and all nontrivial weight spaces of such a module are infinite-dimensional. As a by-product, it is obtained that every simple weight module over Lie algebra of this type with a nontrivial finite-dimensional weight space is a Harish-Chandra module.
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 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The research presented in this paper was carried out during the visit of author to Wilfrid Laurier University, Canada. The author thanks Wilfrid Laurier University for hospitality. The author would like to thank Prof. K. Zhao for stimulating discussion. The author also thanks the referees for good suggestions. This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11271165, 11471333） and the Natural Science Foundation for Colleges and Universities in Jiangsu Province （No. 14KJBll0006）.
文摘There are two extensions of Virasoro algebra with particular importance in superstring theory： the Ramond algebra and the Neveu-Schwarz algebra, which are Z2-graded extensions of the Virasoro algebra. In this paper, we show that the support of a simple weight module over the Ramond algebra with an infinite-dimensional weight space coincides with the weight lattice and that all intersections of non-trivial weight spaces and odd part or even part of the module are infinite-dimensional. This result together with the one that we have obtained over the Neveu-Schwarz algebra generalizes the result for the Virasoro algebra to the super-Virasoro algebras.
文摘It is shown that there are no simple mixed modules over the twisted N = 1 Schrodinger-Neveu-Schwarz algebra, which implies that every irreducible weight module over it with a non-trivial finite-dimensional weight space is a Harish-Chandra module.
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 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The authors would like to thank the referees for nice suggestions. This work was supported in part by the National Natural Science Foundation of China （Grant No. 11301143） and the school fund of Henan University （yqpy20140044）.
文摘For any module V over the two-dimensional non-abelian Lie algebra b and scalar a ∈C, we define a class of weight modules Fα（V） with zero central charge over the affine Lie algebra A（1）. These weight modules have infinite- dimensional weight spaces if and only if V is infinite dimensional. In this paper, we will determine necessary and sufficient conditions for these modules Fα（V） to be irreducible. In this way, we obtain a lot of irreducible weight A1（1）-modules with infinite-dimensional weight spaces.
基金supported by National Natural Science Foundation of China (Grant No.11601167)。
文摘The Heisenberg double D_(q)(E_(2))of the quantum Euclidean group O_(q)(E_(2))is the smash product of O_(q)(E_(2))with its Hopf dual U_(q)(e_(2)).For the algebra D_(q)(E_(2)),explicit descriptions of its prime,primitive and maximal spectra are obtained.All the prime factors of D_(q)(E_(2))are presented as generalized Weyl algebras.As a result,we obtain that the algebra D_(q)(E_(2))has no finite-dimensional representations,and D_(q)(E_(2))cannot have a Hopf algebra structure.The automorphism groups of the quantum Euclidean group and its Heisenberg double are determined.Some centralizers are explicitly described via generators and defining relations.This enables us to give a classification of simple weight modules and the so-called a-weight modules over the algebra D_(q)(E_(2)).
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）.
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.
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.
