量子代数${\cal U}_q(f(K,H))$的量子伴随作用

The aim of this paper is to study the adjoint action for the quantum algebra Uq（f（K, H））, which is a natural generalization of quantum algebra Uq（sl2） and is regarded as a class of generalized Weyl algebra..The structure theorem of its locally finite subalgebra F（Uq（f（K, H））） is given.     

Gelfand–Kirillov Dimensions of the Z~2-graded Oscillator Representations of sl(n)

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.     

$L^{\Phi}(I,X)$中的最佳同时逼近

Let X be a Banach space and Ф be an Orlicz function. Denote by L^Ф（I,X） the space of X-valued （I）-integrable functions on the unit interval I equipped with the Luxemburg norm. For f1,f2,... ,fm ∈ L^Ф（I,X）, a distance formula distv（f1,f2,... ,fm,L^Ф（I,G）） is presented, where G is a close subspace of X. Moreover, some existence and characterization results concerning the best simultaneous approximation of L^Ф （I, G） in L^Ф （I, X） axe given.     

$(\omega'')$性质及其摄动

In this note we define the property （ω′）, a variant of Weyl’s theorem, and establish for a bounded linear operator defined on a Hilbert space the necessary and sufficient conditions for which property （ω′） holds by means of the variant of the essential approximate point spectrum σ1（·） and the spectrum defined in view of the property of consistency in Fredholm and index. In addition, the perturbation of property （ω′） is discussed.     

${\cal U}_q(g)$在其正部分${\cal U}_q^ (g)$上的作用

In this paper, two kinds of skew derivations of a type of Nichols algebras are intro- duced, and then the relationship between them is investigated. In particular they satisfy the quantum Serre relations. Therefore, the algebra generated by these derivations and corresponding automorphisms is a homomorphic image of the Drinfeld-Jimbo quantum enveloping algebra Uq^＋（g）, which proves the Nichols algebra becomes a/gq（g）-module algebra. But the Nichols algebra considered here is exactly Uq^＋（g）, namely, the positive part of the Drinfeld-Jimbo quantum enveloping algebra Uq^＋（g）, it turns out that Uq^＋（g） is aUq^＋（g）-module algebra.     

性质$(\omega)$的注解

In this note we study the property （ω）, a variant of Weyl＇s theorem introduced by Rakocevic, by means of the new spectrum. We establish for a bounded linear operator defined on a Banach space a necessary and sufficient condition for which both property （ω） and approximate Weyl＇s theorem hold. As a consequence of the main result, we study the property （ω） and approximate Weyl＇s theorem for a class of operators which we call the λ-weak-H（p） operators.     

Differential Equations and Singular Vectors in Verma Modules over sl(n,C)

Xu introduced a system of partial differential equations to investigate singular vectors in the Verma module of highest weightλover sl(n,C).He gave a differential-operator representation of the symmetric group S_n on the corresponding space of truncated power series and proved that the solution space of the system is spanned by{σ(1)|σ∈S_n}.It is known that S_n is also the Weyl group of sl(n,C)and generated by all reflections s_αwith positive rootsα.We present an explicit formula of the solution s_α(1)for every positive rootαand show directly that s_α(1)is a polynomial if and only if〈λ+ρ,α〉is a nonnegative integer.From this,we can recover a formula of singular vectors given by Malikov et al..     

区传递的$2-(v, k, 1)$ 设计与单群$E_6(q)$

This article is a contribution to the study of block-transitive automorphism groups of 2-（v,k,1） block designs. Let D be a 2-（v,k,1） design admitting a block-transitive, pointprimitive but not flag-transitive automorphism group G. Let kr = （k,v-1） and q = pf for prime p. In this paper we prove that if G and D are as above and q （3（krk-kr ＋ 1）f）1/3, then G does not admit a simple group E6（q） as its socle.     

Wely定理与$(\omega)$性质的等价性

We call T C B（H） consistent in Fredholm and index （briefly a CFI operator） if for each B ∈ B（H）, TB and BT are Fredholm together and the same index of B, or not Fredholm together. Using a new spectrum defined in view of the CFI operator, we give the equivalence of Weyl＇s theorem and property （ω） for T and its conjugate operator T^＊. In addition, the property （ω） for operator matrices is considered.     

Banach空间中含$(H,\eta)$单调算子变分包含组解的迭代法

In this paper, we introduce and study a new system of variational inclusions involving （H, η）-monotone operators in Banach space. Using the resolveut operator associated with （H, η）- monotone operators, we prove the existence and uniqueness of solutions for this new system of variational inclusions. We also construct a new algorithm for approximating the solution of this system and discuss the convergence of the iterative sequence generated by the algorithm.     

量子群u_q(sl(2))中理想的唯一分解

本文利用量子群uq(sl(2))的表示及其理想的生成子,证明了uq(sl(2))的任一非零理想在某种意义下可唯一分解为若干个素理想的乘积.由此得到理想的求根公式.     

kQ#k<σ>的Clebsch—Gordan问题

主要讨论斜群代数kQ#k<σ>的Clebsch-Gordan问题.特别地,当Q是A_n,D_n型简图,以及Kronecker简图时,给出kQ#k<σ>的Clebsch-Gordan公式.     

Gelfand-Kirillov dimensions of the ℤ2-graded oscillator representations of \mathfrak{s}\mathfrak{l}(n)

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.     

Laguerre functions and representations of su(1,1)

Spectral analysis of a certain doubly infinite Jacobi operator leads to orthogonality relations for confluent hypergeometric functions, which are called Laguerre functions. This doubly infinite Jacobi operator corresponds to the action of a parabolic element of the Lie algebra su(l, 1). The Clebsch-Gordan coefficients for the tensor product representation of a positive and a negative discrete series representation of su(l,l) are determined for the parabolic bases. They turn out to be multiples of Jacobi functions. From the interpretation of Laguerre polynomials and functions as overlap coefficients, we obtain a product formula for the Laguerre polynomials, given by an integral over Laguerre functions, Jacobi functions and continuous dual Hahn polynomials.     

Differential equations and singular vectors in Verma modules over sl(n, ℂ)

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 S_n on the corresponding space of truncated power series and proved that the solution space of the system is spanned by {σ(1) | σ ∈ S_n}. It is known that S_n is also the Weyl group of sl(n,C) and generated by all reflections s_α with positive roots α. We present an explicit formula of the solution s_α(1) for every positive root α and show directly that s_α(1) is a polynomial if and only if <λ + ρ, α> is a nonnegative integer. From this, we can recover a formula of singular vectors given by Malikov et al..     

On Dynamical Adjoint Functor

We give an explicit formula relating the dynamical adjoint functor and dynamical twist over nonalbelian base to the invariant pairing on parabolic Verma modules. As an illustration, we give explicit $U\bigl(\mathfrak{sl}(n)\bigr)$ - and $U_\hbar\bigl(\mathfrak{sl}(n)\bigr)$ -invariant star product on projective spaces.     

sl_2(R)上Harish-Chandra模及其应用

将ｓｌ２（Ｒ）上不可约Ｈａｒｉｓｈ－Ｃｈａｎｄｒａ模及ｓｌ２（Ｒ）上不可分解的Ｈａｒｉｓｈ－Ｃｈａｎｄｒａ模进行了完全分类，得到了与ｓｌ２（Ｃ）上模分类的不同形式．作为应用，又构造了实Ｖｉｒａｓｏｒｏ代数的一类新的不可约表示．     

A combinatorial formula for affine Hall–Littlewood functions via a weighted Brion theorem

We present a new combinatorial formula for Hall–Littlewood functions associated with the affine root system of type \({{\tilde{A}}}_{n-1}\), i.e., corresponding to the affine Lie algebra \({{\widehat{\mathfrak {sl}}}}_n\). Our formula has the form of a sum over the elements of a basis constructed by Feigin, Jimbo, Loktev, Miwa and Mukhin in the corresponding irreducible representation. Our formula can be viewed as a weighted sum of exponentials of integer points in a certain infinite-dimensional convex polyhedron. We derive a weighted version of Brion’s theorem and then apply it to our polyhedron to prove the formula.     

Tensor products on quiver representations

The aim of this article is to translate the well-known tensor product of representations of a group given by diagonal action to the case of representations of a quiver. We provide three different approaches and exhibit their close relationship to the point-wise tensor product, which is considered in [M. Herschend, Solution to the Clebsch-Gordan problem for Kronecker representations. U.U.D.M Project Report 2003:P1, Uppsala University, 2003; M. Herschend, Solution to the Clebsch-Gordan problem for representations of quivers of type , J. Algebra Appl. 4 (5) (2005) 481-488; M. Herschend, On the Clebsch-Gordan problem for quiver representations. U.U.D.M Report 2005:43, Uppsala University, 2005].