In this paper ， we obtain that, if G is either a basic classical Lie superalgebra of I type or G= B(0, n), every Verma module over G has a unique minimal submodule, And this fails for G=F(4).
This paper is concerned with the dimension of the space of the homomorphisms between the Verma modules over a basic classical Lie superalgebra and the kernel of such homomorphism.
This paper deals with the Verma module of quantum group U_(q)(B_(2)) when q is a root of 1.It estimates how many classes of Verma submodules which are not isomorphic to each other are contained in Verma module.The cha...This paper deals with the Verma module of quantum group U_(q)(B_(2)) when q is a root of 1.It estimates how many classes of Verma submodules which are not isomorphic to each other are contained in Verma module.The character of irreducible highest weight module ofU_(q)(B_(2)) has also been given.展开更多
In this paper,we describe the structure of Verma modules over the two kinds of Lie algebras g(λ)of W-type.We determine the reducibility and the singular vectors of their Verma modules under some conditions.
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
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 modu...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
Xu introduced a system of partial differential equations to investigate singular vectors in the Verma module of highest weight λ, over sl（n,C）. He gave a differential-operator representation of the symmetric group ...Xu introduced a system of partial differential equations to investigate singular vectors in the Verma module of highest weight λ, over sl（n,C）. He gave a differential-operator representation of the symmetric group Sn on the corresponding space of truncated power series and proved that the solution space of the system is spanned by {σ（1）｜ σ ∈ Sn}. It is known that Sn is also the Weyl group of sl（n, C） and generated by all reflections sα with positive roots α. We present an explicit formula of the solution sα（1） for every positive root α and show directly that sα（1） is a polynomial if and only if （λ＋p, α） is a nonnegative integer. From this, we can recover a formula of singular vectors given by Malikov et al..展开更多
<正> The q-deformation of Verma theory for the Lie algebra is studied in this paper. Theindecomposable representations and the induced representations of quantum universal envelop-ing algebra sl_q(3) are constru...<正> The q-deformation of Verma theory for the Lie algebra is studied in this paper. Theindecomposable representations and the induced representations of quantum universal envelop-ing algebra sl_q(3) are constructed on the q-deformed Verma space and the quotient spacesrespectively. We put stress on the discussion of the case in which q is a root of unity. Usingthe new representation constrained in the subalgebra sl_q(2), we systematically constructthe new series of solutions (Yang-Baxter matrices) for Yang-Baxter equation without spectralparameter.展开更多
Given a suitable ordering of the positive root system associated with a semisimple Lie algebra,there exists a natural correspondence between Verma modules and related polynomial algebras. With this, the Lie algebra ac...Given a suitable ordering of the positive root system associated with a semisimple Lie algebra,there exists a natural correspondence between Verma modules and related polynomial algebras. With this, the Lie algebra action on a Verma module can be interpreted as a differential operator action on polynomials, and thus on the corresponding truncated formal power series. We prove that the space of truncated formal power series gives a differential-operator representation of the Weyl group W. We also introduce a system of partial differential equations to investigate singular vectors in the Verma module. It is shown that the solution space of the system in the space of truncated formal power series is the span of {w(1) | w ∈ W }. Those w(1) that are polynomials correspond to singular vectors in the Verma module. This elementary approach by partial differential equations also gives a new proof of the well-known BGG-Verma theorem.展开更多
Over a field of characteristic p>0,the first cohomology of the orthogonal symplectic Lie superalgebra osp(1,2)with coefficients in baby Verma modules and simple modules is determined by use of the weight space deco...Over a field of characteristic p>0,the first cohomology of the orthogonal symplectic Lie superalgebra osp(1,2)with coefficients in baby Verma modules and simple modules is determined by use of the weight space decompositions of these modules relative to a Cartan subalgebra of osp(1,2).As a byproduct,the first cohomology of osp(1,2)with coefficients in the restricted enveloping algebra(under the adjoint action)is not trivial.展开更多
Let■be a compatible total order on the additive group Z^2,and L be the rank two HeisenbergVirasoro algebra.For any c=(c1,c2,c3,c4)∈C^4,we define a Z^2-graded Verma module M(c,■)for L.A necessary and sufficient cond...Let■be a compatible total order on the additive group Z^2,and L be the rank two HeisenbergVirasoro algebra.For any c=(c1,c2,c3,c4)∈C^4,we define a Z^2-graded Verma module M(c,■)for L.A necessary and sufficient condition for M(c,■)to be irreducible is provided.Moreover,the maximal Z^2-graded submodules of M(c,■)are characterized when M(c,■)is reducible.展开更多
文摘In this paper ， we obtain that, if G is either a basic classical Lie superalgebra of I type or G= B(0, n), every Verma module over G has a unique minimal submodule, And this fails for G=F(4).
文摘This paper is concerned with the dimension of the space of the homomorphisms between the Verma modules over a basic classical Lie superalgebra and the kernel of such homomorphism.
文摘This paper deals with the Verma module of quantum group U_(q)(B_(2)) when q is a root of 1.It estimates how many classes of Verma submodules which are not isomorphic to each other are contained in Verma module.The character of irreducible highest weight module ofU_(q)(B_(2)) has also been given.
基金Supported by the National Natural Science Foundation of China(Grant No.11771122).
文摘In this paper,we describe the structure of Verma modules over the two kinds of Lie algebras g(λ)of W-type.We determine the reducibility and the singular vectors of their Verma modules under some conditions.
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.
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.
文摘Xu introduced a system of partial differential equations to investigate singular vectors in the Verma module of highest weight λ, over sl（n,C）. He gave a differential-operator representation of the symmetric group Sn on the corresponding space of truncated power series and proved that the solution space of the system is spanned by {σ（1）｜ σ ∈ Sn}. It is known that Sn is also the Weyl group of sl（n, C） and generated by all reflections sα with positive roots α. We present an explicit formula of the solution sα（1） for every positive root α and show directly that sα（1） is a polynomial if and only if （λ＋p, α） is a nonnegative integer. From this, we can recover a formula of singular vectors given by Malikov et al..
基金Project supported in part by the National Natural Science Foundation of China.
文摘<正> The q-deformation of Verma theory for the Lie algebra is studied in this paper. Theindecomposable representations and the induced representations of quantum universal envelop-ing algebra sl_q(3) are constructed on the q-deformed Verma space and the quotient spacesrespectively. We put stress on the discussion of the case in which q is a root of unity. Usingthe new representation constrained in the subalgebra sl_q(2), we systematically constructthe new series of solutions (Yang-Baxter matrices) for Yang-Baxter equation without spectralparameter.
基金supported by National Natural Science Foundation of China(Grant No.11326059)
文摘Given a suitable ordering of the positive root system associated with a semisimple Lie algebra,there exists a natural correspondence between Verma modules and related polynomial algebras. With this, the Lie algebra action on a Verma module can be interpreted as a differential operator action on polynomials, and thus on the corresponding truncated formal power series. We prove that the space of truncated formal power series gives a differential-operator representation of the Weyl group W. We also introduce a system of partial differential equations to investigate singular vectors in the Verma module. It is shown that the solution space of the system in the space of truncated formal power series is the span of {w(1) | w ∈ W }. Those w(1) that are polynomials correspond to singular vectors in the Verma module. This elementary approach by partial differential equations also gives a new proof of the well-known BGG-Verma theorem.
基金This work is supported by Heilongjiang Provincial Natural Science Foundation of China(YQ2020A005)Natural Science Foundation of China(12061029).
文摘Over a field of characteristic p>0,the first cohomology of the orthogonal symplectic Lie superalgebra osp(1,2)with coefficients in baby Verma modules and simple modules is determined by use of the weight space decompositions of these modules relative to a Cartan subalgebra of osp(1,2).As a byproduct,the first cohomology of osp(1,2)with coefficients in the restricted enveloping algebra(under the adjoint action)is not trivial.
基金supported by National Natural Science Foundation of China(Grant Nos.11471268 and 11531004)。
文摘Let■be a compatible total order on the additive group Z^2,and L be the rank two HeisenbergVirasoro algebra.For any c=(c1,c2,c3,c4)∈C^4,we define a Z^2-graded Verma module M(c,■)for L.A necessary and sufficient condition for M(c,■)to be irreducible is provided.Moreover,the maximal Z^2-graded submodules of M(c,■)are characterized when M(c,■)is reducible.
