In this article,by studying the Bernstein degrees and Goldie rank polynomials,we es-tablish a comparison between the irreducible representations of G=GL_(n)(C)possessing the minimal Gelfand-Kirillov dimension and thos...In this article,by studying the Bernstein degrees and Goldie rank polynomials,we es-tablish a comparison between the irreducible representations of G=GL_(n)(C)possessing the minimal Gelfand-Kirillov dimension and those induced from finite-dimensional representations of the maximal parabolic subgroup of G of type(n-1,1).We give the transition matrix between the two bases for the corresponding coherent families.展开更多
Let g be a classical complex simple Lie algebra and q be a parabolic subalgebra.Let M be a generalized Verma module induced from a one dimensional representation of q.Such M is called a scalar generalized Verma module...Let g be a classical complex simple Lie algebra and q be a parabolic subalgebra.Let M be a generalized Verma module induced from a one dimensional representation of q.Such M is called a scalar generalized Verma module.In this paper,we will determine the reducibility of scalar generalized Verma modules associated to maximal parabolic subalgebras by computing explicitly the Gelfand-Kirillov dimension of the corresponding highest weight modules.展开更多
Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension...Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method： Grobner-Shirshov basis method. We develop the GrSbner-Shirshov basis theory of differential difference al- gebras, and of finitely generated modules over differential difference algebras, respectively. Then, via GrSbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.展开更多
We study the growth and the Gelfand-Kirillov dimension(GK-dimension)of the generalized Weyl algebra(GWA)A=D(σ,a),where D is a polynomial algebra or a Laurent polynomial algebra.Several necessary and sufficient condit...We study the growth and the Gelfand-Kirillov dimension(GK-dimension)of the generalized Weyl algebra(GWA)A=D(σ,a),where D is a polynomial algebra or a Laurent polynomial algebra.Several necessary and sufficient conditions for GKdim(A)=GKdim(D)+1 are given.In particular,we prove a dichotomy of the GK-dimension of GWAs over the polynomial algebra in two indeterminates,i.e.,GKdim(A)is either 3 or∞in this case.Our results generalize several existing results in the literature and can be applied to determine the growth,GK-dimension,simplicity and cancellation properties of some GWAs.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
The Gelfand–Kirillov dimension is an invariant which can measure the size of infinitedimensional algebraic structures. In this article, we show that it can also measure the reducibility of scalar generalized Verma mo...The Gelfand–Kirillov dimension is an invariant which can measure the size of infinitedimensional algebraic structures. In this article, we show that it can also measure the reducibility of scalar generalized Verma modules. In particular, we use it to determine the reducibility of scalar generalized Verma modules associated with maximal parabolic subalgebras in the Hermitian symmetric case.展开更多
We find an exact formula of Gelfand-Kirillov dimensions for the infinite-dimensional explicit irreducible sl（n,F）-modules that appeared in the Z2-graded oscillator generalizations of the classical theorem on harmoni...We find an exact formula of Gelfand-Kirillov dimensions for the infinite-dimensional explicit irreducible sl（n,F）-modules that appeared in the Z2-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. Three infinite subfamilies of these modules have the minimal Gelfand-Kirillov dimension. They contain weight modules with unbounded weight multiplicities and completely pointed modules.Service E-mail this articleAdd to my bookshelfAdd to citation managerE-mail AlertRSSArticles by authors展开更多
基金Z.Bai was supported in part by the National Natural Science Foundation of China(Grant No.12171344)the National Key R&D Program of China(Grant Nos.2018YFA0701700 and 2018YFA0701701)+5 种基金Y.Chen was supported in part by the National Natural Science Foundation of China(Grant No.12301035)the Natural Science Foundation of Jiangsu Province(Grant No.BK20221057)D.Liu was supported by National Key R&D Program of China(Grant No.2022YFA1005300)the National Natural Science Foundation of China(Grant No.12171421)B.Sun was supported by National Key R&D Program of China(Grant Nos.2022YFA1005300 and 2020YFA0712600)New Cornerstone Investigator Program。
文摘In this article,by studying the Bernstein degrees and Goldie rank polynomials,we es-tablish a comparison between the irreducible representations of G=GL_(n)(C)possessing the minimal Gelfand-Kirillov dimension and those induced from finite-dimensional representations of the maximal parabolic subgroup of G of type(n-1,1).We give the transition matrix between the two bases for the corresponding coherent families.
基金Supported by the National 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 classical complex simple Lie algebra and q be a parabolic subalgebra.Let M be a generalized Verma module induced from a one dimensional representation of q.Such M is called a scalar generalized Verma module.In this paper,we will determine the reducibility of scalar generalized Verma modules associated to maximal parabolic subalgebras by computing explicitly the Gelfand-Kirillov dimension of the corresponding highest weight modules.
文摘Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method： Grobner-Shirshov basis method. We develop the GrSbner-Shirshov basis theory of differential difference al- gebras, and of finitely generated modules over differential difference algebras, respectively. Then, via GrSbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.
基金supported by Huizhou University(Grant Nos.hzu202001 and 2021JB022)the Guangdong Provincial Department of Education(Grant Nos.2020KTSCX145 and 2021ZDJS080)。
文摘We study the growth and the Gelfand-Kirillov dimension(GK-dimension)of the generalized Weyl algebra(GWA)A=D(σ,a),where D is a polynomial algebra or a Laurent polynomial algebra.Several necessary and sufficient conditions for GKdim(A)=GKdim(D)+1 are given.In particular,we prove a dichotomy of the GK-dimension of GWAs over the polynomial algebra in two indeterminates,i.e.,GKdim(A)is either 3 or∞in this case.Our results generalize several existing results in the literature and can be applied to determine the growth,GK-dimension,simplicity and cancellation properties of some GWAs.
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 National Science Foundation of China(Grant No.11601394)supported by the National Science Foundation of China(Grant No.11701381)Guangdong Natural Science Foundation(Grant No.2017A030310138)
文摘The Gelfand–Kirillov dimension is an invariant which can measure the size of infinitedimensional algebraic structures. In this article, we show that it can also measure the reducibility of scalar generalized Verma modules. In particular, we use it to determine the reducibility of scalar generalized Verma modules associated with maximal parabolic subalgebras in the Hermitian symmetric case.
基金Supported by National Natural Science Foundation of China(Grant No.11171324)
文摘We find an exact formula of Gelfand-Kirillov dimensions for the infinite-dimensional explicit irreducible sl（n,F）-modules that appeared in the Z2-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. Three infinite subfamilies of these modules have the minimal Gelfand-Kirillov dimension. They contain weight modules with unbounded weight multiplicities and completely pointed modules.Service E-mail this articleAdd to my bookshelfAdd to citation managerE-mail AlertRSSArticles by authors
