In this paper, the representation theory for the arlene Lie algebra H4 associated to the Nappi-Witten Lie algebra H4 is studied. Polynomial representations of the affine Nappi-Witten Lie algebra H4 are given.
In this paper, we discuss the pairing problem of generators in four affine Lie algebra. That is, for any given imaginary root vector x∈g(A) , there exists y such that x and y generate a subalgebra cont...In this paper, we discuss the pairing problem of generators in four affine Lie algebra. That is, for any given imaginary root vector x∈g(A) , there exists y such that x and y generate a subalgebra containing g′(A).展开更多
For any finite-dimensional semisimple Lie algebra g, a Z＋-graded vertex algebra is construsted on the vacuum representation Vk（g[θ]of g[θ]）,which is a one-dimentionM central extension of 8-invariant subspace on t...For any finite-dimensional semisimple Lie algebra g, a Z＋-graded vertex algebra is construsted on the vacuum representation Vk（g[θ]of g[θ]）,which is a one-dimentionM central extension of 8-invariant subspace on the loop algebra Lg=g C（（t^1/p））.展开更多
Non-commutative Poisson algebras are the algebras having both an associa- tive algebra structure and a Lie algebra structure together with the Leibniz law. In this paper, the non-commutative poisson algebra structures...Non-commutative Poisson algebras are the algebras having both an associa- tive algebra structure and a Lie algebra structure together with the Leibniz law. In this paper, the non-commutative poisson algebra structures on the Lie algebras sln（Cq^-） are determined.展开更多
Let k be a field and q a nonzero element in k such that the square roots of q are in k. We use Hq to denote an affne Hecke algebra over k of type G2 with parameter q. The purpose of this paper is to study representati...Let k be a field and q a nonzero element in k such that the square roots of q are in k. We use Hq to denote an affne Hecke algebra over k of type G2 with parameter q. The purpose of this paper is to study representations of Hq by using based rings of two-sided cells of an affne Weyl group W of type G2. We shall give the classification of irreducible representations of Hq. We also remark that a calculation in [11] actually shows that Theorem 2 in [1] needs a modification, a fact is known to Grojnowski and Tanisaki long time ago. In this paper we also show an interesting relation between Hq and an Hecke algebra corresponding to a certain Coxeter group. Apparently the idea in this paper works for all affne Weyl groups, but that is the theme of another paper.展开更多
For the extended affine Lie algebra gl_(l+1)(Cq),we give a sufficient and necessary condition in this article to guarantee that the gl_(l+1)(Cq)-module V constructed by using free field realizaion is unitarizable.
We conjecture an explicit bound on the prime characteristic of a field, under which the Weyl modules of affine sl_2 and the minimal series modules of Virasoro algebra remain irreducible, and Goddard-Kent-Olive coset c...We conjecture an explicit bound on the prime characteristic of a field, under which the Weyl modules of affine sl_2 and the minimal series modules of Virasoro algebra remain irreducible, and Goddard-Kent-Olive coset construction for affine sl_2 is valid.展开更多
In this paper, we categorify the algebra Uq（sl2） with the same approach as in [A. Lauda, Adv. Math. （2010）, arXiv：math.QA/0803.3662; M. Khovanov, Comm. Algebra 11 （2001） 5033]. The algebra U =Uq（sl2） is obtai...In this paper, we categorify the algebra Uq（sl2） with the same approach as in [A. Lauda, Adv. Math. （2010）, arXiv：math.QA/0803.3662; M. Khovanov, Comm. Algebra 11 （2001） 5033]. The algebra U =Uq（sl2） is obtained from Uq（sl2） by adjoining a collection of orthogonal idempotents 1λ,λ ∈ P, in which P is the weight lattice of Uq（sl2）. Under such construction the algebra U is decomposed into a direct sum λ∈p 1λ,U1λ. We set the collection of λ∈ P as the objects of the category U, 1-morphisms from λ to λ′ are given by 1λ,U1λ, and 2-morphisms are constructed by some semilinear form defined on U. Hence we get a 2-category u from the algebra Uq（sl2）.展开更多
The purpose of this paper is to construct quotient algebras L(A)1C/I(A)of complex degenerate composition Lie algebras L(A)1C by some ideals,where L(A)1C is defined via Hall algebras of tubular algebras A,and to prove ...The purpose of this paper is to construct quotient algebras L(A)1C/I(A)of complex degenerate composition Lie algebras L(A)1C by some ideals,where L(A)1C is defined via Hall algebras of tubular algebras A,and to prove that the quotient algebras L(A)1C/I(A)are isomorphic to the corresponding affine Kac-Moody algebras.Moreover,it is shown that the Lie algebra Lre(A)1C generated by A-modules with a real root coincides with the degenerate composition Lie algebra L(A)1C generated by simple A-modules.展开更多
This paper discusses two problems:one is some important theories and algorithms of affine bracket algebra;the other is about their applications in mechanical theorem proving.First we give some efficient algorithms inc...This paper discusses two problems:one is some important theories and algorithms of affine bracket algebra;the other is about their applications in mechanical theorem proving.First we give some efficient algorithms including the boundary expanding algorithm which is a key feature in application.We analyze the characteristics of the boundary operator and this is the base for the implementation of the system.We also give some new theories or methods about the exact division,the representations and structure of affine geometry and so on.In practice,we implement the mechanical auto-proving system in Maple 10 based on the above algorithms and theories.Also we test about more than 100 examples and compare the results with the methods before.展开更多
In this paper, the extended affine Lie algebra sl2（Cq） is quantized from three different points of view, which produces three non-commutative and non-cocommutative Hopf algebra structures, and yields other three qua...In this paper, the extended affine Lie algebra sl2（Cq） is quantized from three different points of view, which produces three non-commutative and non-cocommutative Hopf algebra structures, and yields other three quantizations by an isomorphism of sl2 （Cq） correspondingly. Moreover, two of these quantizations can be restricted to the extended affine Lie algebra sl2（Cq）.展开更多
Imaginary Verma modules, parabolic imaginary Verma modules, and Verma modules at level zero for double affine Lie algebras are constructed using three different triangular decompositions. Their relations are investiga...Imaginary Verma modules, parabolic imaginary Verma modules, and Verma modules at level zero for double affine Lie algebras are constructed using three different triangular decompositions. Their relations are investigated, and several results are generalized from the affine Lie algebras. In particular, imaginary highest weight modules, integrable modules, and irreducibility criterion are also studied.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
Some properties of automorphisms of affine Lie algebras are presented.Automorphisms with order three are classified up to conjugacy,and their fixed point sets are also computed.As a corollary,it is proved that two aut...Some properties of automorphisms of affine Lie algebras are presented.Automorphisms with order three are classified up to conjugacy,and their fixed point sets are also computed.As a corollary,it is proved that two automorphisms of order three are conjugate if and only if their fixed point sets are isomorphic.展开更多
level over module or We obtain that every irreducible quasifinite module with non-zero the twisted affine Nappi-Witten algebra is either a highest weight a lowest one
Let g be a(twisted or untwisted)affine Kac-Moody algebra,and μ be a diagram automorphism of g.In this paper,we give an explicit realization for the universal central extensionˆg[μ]of the twisted loop algebra of g wi...Let g be a(twisted or untwisted)affine Kac-Moody algebra,and μ be a diagram automorphism of g.In this paper,we give an explicit realization for the universal central extensionˆg[μ]of the twisted loop algebra of g with respect toμ,which provides a Moody-Rao-Yokonuma presentation for the algebraˆg[μ]whenμis non-transitive,and the presentation is indeed related to the quantization of twisted toroidal Lie algebras.展开更多
We construct a class of modules for extended affine Lie algebra gl l（Cq） by using the free fields. A necessary and sufficient condition is given for those modules being irreducible.
基金Supported in part by NSFC(10871125,10931006)a grant of Science and Technology Commission of Shanghai Municipality(09XD1402500)
文摘In this paper, the representation theory for the arlene Lie algebra H4 associated to the Nappi-Witten Lie algebra H4 is studied. Polynomial representations of the affine Nappi-Witten Lie algebra H4 are given.
文摘In this paper, we discuss the pairing problem of generators in four affine Lie algebra. That is, for any given imaginary root vector x∈g(A) , there exists y such that x and y generate a subalgebra containing g′(A).
基金Supported in part by National Natural Science Foundation of China under Grant No. 10971071the Outstanding Youth Fund of Henan Province under Grant No. 0512000100Innovation Fund of Colleges and Universities in Henan Province
文摘In this paper, we construct a new algebra structure 7-twisted atone Lie algebra sl（3,C）[θ] and study its vertex operator representations.
基金National Key Basic Research Project of China under Grant Nos.2004CB318000 and 2006CB805905National Natural Science Foundation of China under Grant No.10471034+1 种基金the Outstanding Youth Fund of Henan Province under Grant No.0512000100Innovation Fund of Colleges and Universities in Henan Province
文摘For any finite-dimensional semisimple Lie algebra g, a Z＋-graded vertex algebra is construsted on the vacuum representation Vk（g[θ]of g[θ]）,which is a one-dimentionM central extension of 8-invariant subspace on the loop algebra Lg=g C（（t^1/p））.
基金supported by NSF of China(11071187)Innovation Program of Shanghai Municipal Education Commission(09YZ336)
文摘Non-commutative Poisson algebras are the algebras having both an associa- tive algebra structure and a Lie algebra structure together with the Leibniz law. In this paper, the non-commutative poisson algebra structures on the Lie algebras sln（Cq^-） are determined.
基金partially supported by Natural Sciences Foundation of China (10671193)
文摘Let k be a field and q a nonzero element in k such that the square roots of q are in k. We use Hq to denote an affne Hecke algebra over k of type G2 with parameter q. The purpose of this paper is to study representations of Hq by using based rings of two-sided cells of an affne Weyl group W of type G2. We shall give the classification of irreducible representations of Hq. We also remark that a calculation in [11] actually shows that Theorem 2 in [1] needs a modification, a fact is known to Grojnowski and Tanisaki long time ago. In this paper we also show an interesting relation between Hq and an Hecke algebra corresponding to a certain Coxeter group. Apparently the idea in this paper works for all affne Weyl groups, but that is the theme of another paper.
基金supported by the NSF of China(11931009,12161141001,12171132 and 11771410)Anhui Initiative in Quantum Information Technologies(AHY150000).
文摘For the extended affine Lie algebra gl_(l+1)(Cq),we give a sufficient and necessary condition in this article to guarantee that the gl_(l+1)(Cq)-module V constructed by using free field realizaion is unitarizable.
基金supported by the National Science Foundation of USA(Grant No. DMS1405131)
文摘We conjecture an explicit bound on the prime characteristic of a field, under which the Weyl modules of affine sl_2 and the minimal series modules of Virasoro algebra remain irreducible, and Goddard-Kent-Olive coset construction for affine sl_2 is valid.
基金Supported by National Natural Science Foundation of China under Grant Nos. 10975102, 10871135, 11031005, and 10871227
文摘In this paper, we categorify the algebra Uq（sl2） with the same approach as in [A. Lauda, Adv. Math. （2010）, arXiv：math.QA/0803.3662; M. Khovanov, Comm. Algebra 11 （2001） 5033]. The algebra U =Uq（sl2） is obtained from Uq（sl2） by adjoining a collection of orthogonal idempotents 1λ,λ ∈ P, in which P is the weight lattice of Uq（sl2）. Under such construction the algebra U is decomposed into a direct sum λ∈p 1λ,U1λ. We set the collection of λ∈ P as the objects of the category U, 1-morphisms from λ to λ′ are given by 1λ,U1λ, and 2-morphisms are constructed by some semilinear form defined on U. Hence we get a 2-category u from the algebra Uq（sl2）.
基金supported in part by the National Natural Science Foundation of China(Grant No.10471034)Famous Youth Foundation of Henan Province(Grant No.0512000100)the Natural Science Foundation of Educational Committee of Henan Province(Grant No.2000110010).
文摘In this paper,we define a P-twisted affine Lie algebra,and construct its realizations by twisted vertex operators.
基金Supported by the National Natural Science Foundation of China(Grant Nos.10371101 and 10671161)
文摘The purpose of this paper is to construct quotient algebras L(A)1C/I(A)of complex degenerate composition Lie algebras L(A)1C by some ideals,where L(A)1C is defined via Hall algebras of tubular algebras A,and to prove that the quotient algebras L(A)1C/I(A)are isomorphic to the corresponding affine Kac-Moody algebras.Moreover,it is shown that the Lie algebra Lre(A)1C generated by A-modules with a real root coincides with the degenerate composition Lie algebra L(A)1C generated by simple A-modules.
基金supported by the National Natural Science Foundation of China(Grant No.10471143)
文摘This paper discusses two problems:one is some important theories and algorithms of affine bracket algebra;the other is about their applications in mechanical theorem proving.First we give some efficient algorithms including the boundary expanding algorithm which is a key feature in application.We analyze the characteristics of the boundary operator and this is the base for the implementation of the system.We also give some new theories or methods about the exact division,the representations and structure of affine geometry and so on.In practice,we implement the mechanical auto-proving system in Maple 10 based on the above algorithms and theories.Also we test about more than 100 examples and compare the results with the methods before.
文摘In this paper, the extended affine Lie algebra sl2（Cq） is quantized from three different points of view, which produces three non-commutative and non-cocommutative Hopf algebra structures, and yields other three quantizations by an isomorphism of sl2 （Cq） correspondingly. Moreover, two of these quantizations can be restricted to the extended affine Lie algebra sl2（Cq）.
基金N. Jing＇s work was partially supported by the Simons Foundation （Grant No. 198129） and the National Natural Science Foundation of China （Grant No. 11271138）, and he also acknowledged the hospitality of Max-Planck Institute for Mathematics in the Sciences at Leipzig during this work.
文摘Imaginary Verma modules, parabolic imaginary Verma modules, and Verma modules at level zero for double affine Lie algebras are constructed using three different triangular decompositions. Their relations are investigated, and several results are generalized from the affine Lie algebras. In particular, imaginary highest weight modules, integrable modules, and irreducibility criterion are also studied.
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(Grant No.19731004)theNatural Science Foundation of Hebei Province.
文摘Some properties of automorphisms of affine Lie algebras are presented.Automorphisms with order three are classified up to conjugacy,and their fixed point sets are also computed.As a corollary,it is proved that two automorphisms of order three are conjugate if and only if their fixed point sets are isomorphic.
基金This work was supported in part by the National Natural Science Foundation of China （Grant No. 11426191）.
文摘level over module or We obtain that every irreducible quasifinite module with non-zero the twisted affine Nappi-Witten algebra is either a highest weight a lowest one
基金supported by National Natural Science Foundation of China(Grant Nos.11531004 and 11701183)the Fundamental Research Funds for the Central Universities(Grant No.20720190069)the Simons Foundation(Grant No.198129)。
文摘Let g be a(twisted or untwisted)affine Kac-Moody algebra,and μ be a diagram automorphism of g.In this paper,we give an explicit realization for the universal central extensionˆg[μ]of the twisted loop algebra of g with respect toμ,which provides a Moody-Rao-Yokonuma presentation for the algebraˆg[μ]whenμis non-transitive,and the presentation is indeed related to the quantization of twisted toroidal Lie algebras.
基金supported by National Natural Science Foundation of China (Grant Nos.10726014, 10801010)
文摘We construct a class of modules for extended affine Lie algebra gl l（Cq） by using the free fields. A necessary and sufficient condition is given for those modules being irreducible.
