系数在模李超代数~$W(m,3,\underline{1})$ 上的~$\frak{gl}(2,\mathbb{F})$ 的一维上同调

研究了系数在模李超代数~$W(m,3,\underline{1})$ 上的~$\frak{gl}(2,\mathbb{F})$ 的一维上同调, 其中~$\mathbb{F}$ 是一个素特征的代数闭域且~$\frak{gl}(2,\mathbb{F})$ 是系数在~$\mathbb{F}$ 上的~$2\times 2$ 阶矩阵李代数. 计算出所有~$\frak{gl}(2,\mathbb{F})$ 到模李超代数~$W(m,3,\underline{1})$ 的子模的导子和内导子. 从而一维上同调~$\textrm{H}^{1}(\frak{gl}(2,\mathbb{F}),W(m,3,\underline{1}))$ 可以完全用矩阵的形式表示.     

Locally finite Lie algebras with root decomposition

Let K be an algebraically closed field of characteristic zero, $\frak {g}$ be a countably dimensional locally finite Lie algebra over K, and $\frak {h} \subset \frak {g}$ be a (a priori non-abelian) locally nilpotent subalgebra of $\frak {g}$ which coincides with its zero Fitting component. We classify all such pairs $(\frak {g}, \frak {h})$ under the assumptions that the locally solvable radical of $\frak {g}$ equals zero and that $\frak {g}$ admits a root decomposition with respect to $\frak {h}$. More precisely, we prove that $\frak {g}$ is the union of reductive subalgebras $\frak {g}_n$ such that the intersections $\frak {g}_n \cap \frak {h}$ are nested Cartan subalgebras of $\frak {g}_n$ with compatible root decompositions. This implies that $\frak {g}$ is root-reductive and that $\frak {h}$ is abelian. Root-reductive locally finite Lie algebras are classified in [6]. The result of the present note is a more general version of the main classification theorem in [9] and is at the same time a new criterion for a locally finite Lie algebra to be root-reductive. Finally we give an explicit example of an abelian selfnormalizing subalgebra $\frak {h}$ of $\frak {g} = \frak {sl}(\infty)$ with respect to which $\frak {g}$ does not admit a root decomposition.Work Supported in Part by the University of Hamburg and the Max Planck Institute for Mathematics, Bonn     

An example of a bireflectional spin group

Let $\frak n$ be the anisotropic norm of a Cayley algebra $\frak C$ over a field F of characteristic different from 2 where -1 is a square. Let Spin$(\frak C, \frak n)$ be the spin group of the quadratic form $\frak n$. We prove that every element in Spin$(\frak C, \frak n)$ is a product of two involutory elements, i.e. Spin$(\frak C, \frak n)$ is birefiectional. Received: 28 February 2002     

A combinatorial proof of a Weyl type formula for hook Schur polynomials

In this paper, we present a simple combinatorial proof of a Weyl type formula for hook Schur polynomials, which was obtained previously by other people using a Kostant type cohomology formula for . In general, we can obtain in a combinatorial way a Weyl type character formula for various irreducible highest weight representations of a Lie superalgebra, which together with a general linear algebra forms a Howe dual pair. This research was supported by 2007 research fund of University of Seoul.     

General Hypergeometric Functions of Matrices and their Connection to Representations of Linear Groups and Lie Algebras

One considers Gelfand’s hypergeometric functions on the space of p×q matrices and their generalizations to the case of multi-dimensional matrices of arbitrary order k ₁×???×k _p. It is shown that these functions form bases of some $\frak g$ -modules, where $\frak g=\frak{gl}(p,\mathbb{C})\times\frak{gl}(q,\mathbb{C})$ or $\frak g=\frak{gl}(k_{1},\mathbb{C})\times\cdots\times\frak{gl}(k_{p},\mathbb{C})$ , respectively.     

A Converse Hawking-Unruh Effect and dS<Superscript>2</Superscript>/CFT Correspondence

Given a local quantum field theory net $ \mathcal{A} $ on the de Sitter spacetime dS ^d , where geodesic observers are thermalized at Gibbons-Hawking temperature, we look for observers that feel to be in a ground state, i.e., particle evolutions with positive generator, providing a sort of converse to the Hawking-Unruh effect. Such positive energy evolutions always exist as noncommutative flows, but have only a partial geometric meaning, yet they map localized observables into localized observables. We characterize the local conformal nets on dS ^d . Only in this case our positive energy evolutions have a complete geometrical meaning. We show that each net has a unique maximal expected conformal subnet, where our evolutions are thus geometrical. In the two-dimensional case, we construct a holographic one-to-one correspondence between local nets $ \mathcal{A} $ on dS ² and local conformal non-isotonic families (pseudonets) $ \mathcal{B} $ on S ¹. The pseudonet $ \mathcal{B} $ gives rise to two local conformal nets $ \mathcal{B}_\pm $ on S ¹, that correspond to the $ \frak{H}_\pm $ horizon components of $ \mathcal{A} $, and to the chiral components of the maximal conformal subnet of $ \mathcal{A} $. In particular, $ \mathcal{A} $ is holographically reconstructed by a single horizon component, namely the pseudonet is a net, iff the translations on $ \frak{H}_\pm $ have positive energy and the translations on $ \frak{H}_\mp $ are trivial. This is the case iff the one-parameter unitary group implementing rotations on dS ² has positive/negative generator. Communicated by Klaus Fredenhagen submitted 07/02/03, accepted: 07/07/03     

2-极大子群的$F_s$-拟正规性

$F$是一个群系. $G$的子群$H$在$G$中称为$F_s$-拟正规的,如果存在$G$的正规子群$T$,使得$HT$在$G$中是$s$-置换的并且$(H\cap T)H_G/H_G$包含在$G/H_G$的$F$超中心$Z^F_\infty(G/H_G)$中.本文利用$F_s$-拟正规子群研究了有限群的结构.获得了某些新的结果.     

Wild ramification in number field extensions of prime degree

We show that if L/ K is a degree p extension of number fields which is wildly ramified at a prime ${\frak p}$ of K of residue characteristic p, then the ramification groups of ${\frak p}$ (in the splitting field of L over K) are uniquely determined by the ${\frak p}$-adic valuation of the discriminant of L /K.Received: 3 July 2002     

Constructions of Metric $(n+1)$-Lie Algebras

Metric n-Lie algebras have wide applications in mathematics and mathematical physics. In this paper, the authors introduce two methods to construct metric (n+1)-Lie algebras from metric n-Lie algebras for n≥2. For a given m-dimensional metric n-Lie algebra(g, [, ···, ], B_g), via one and two dimensional extensions ￡=g+IFc and g0= g+IFx~(-1)+IFx~0 of the vector space g and a certain linear function f on g, we construct(m+1)-and (m+2)-dimensional (n+1)-Lie algebras(￡, [, ···, ]cf) and(g0, [, ···, ]1), respectively.Furthermore, if the center Z(g) is non-isotropic, then we obtain metric(n + 1)-Lie algebras(L, [, ···, ]cf, B) and(g0, [, ···, ]1, B) which satisfy B|g×g = Bg. Following this approach the extensions of all(n + 2)-dimensional metric n-Lie algebras are discussed.     

Newton's formula for

This paper presents an explicit relation between the two sets which are well-known generators of the center of the universal enveloping algebra of the Lie algebra : one by Capelli (1890) and the other by Gelfand (1950). Our formula is motivated to give an exact analogy for the classical Newton's formula connecting the elementary symmetric functions and the power sum symmetric functions. The formula itself can be deduced from a more general result on Yangians obtained by Nazarov. Our proof is elementary and has an advantage in its direct accessibility.

     

Between\widehat{gl}(\infty ) and\widehat{sl}_N affine algebras. I. Geometrical actions

We consider the central extended $\widehat{gl}(\infty )$ Lie algebra and a set of its subalgebras parametrized by |q|=1, which coincides with the embedding of the quantum tori Lie algebras (QTLA) in $\widehat{gl}(\infty )$ . Forq ^N=1 there exists an ideal, and a factor over this ideal is isomorphic to an $\widehat{sl}_{N(z)} $ affine algebra. For a generic valueq the corresponding subalgebras are dense in $\widehat{gl}(\infty )$ . Thus, they interpolate between $\widehat{gl}(\infty )$ and $\widehat{sl}_{N(z)} $ . All these subalgebras are fixed points of automorphism of $\widehat{gl}(\infty )$ . Using the automorphisms, we construct geometrical actions for the subalgebras, starting from the Kirillov-Kostant form and the corresponding geometrical action for $\widehat{gl}(\infty )$ .     

On pure subgroups of locally compact abelian groups

In this note, we construct an example of a locally compact abelian group G = C × D (where C is a compact group and D is a discrete group) and a closed pure subgroup of G having nonpure annihilator in the Pontrjagin dual $\hat{G}$, answering a question raised by Hartman and Hulanicki. A simple proof of the following result is given: Suppose ${\frak K}$ is a class of locally compact abelian groups such that $G \in {\frak K}$ implies that $\hat{G} \in {\frak K}$ and nG is closed in G for each positive integer n. If H is a closed subgroup of a group $G \in {\frak K}$, then H is topologically pure in G exactly if the annihilator of H is topologically pure in $\hat{G}$. This result extends a theorem of Hartman and Hulanicki.Received: 4 April 2002     

Cocycle Perturbation on Banach Algebras

Let $α$ be a flow on a Banach algebra$\mathcal{B}$, and $t → u_t$ a continuous function from$\boldsymbol{R}$into the group of invertible elements of$\mathcal{B}$such that $u_sα_s(u_t) = u_{s+t}, s, t ∈ \boldsymbol{R}$. Then $β_t$ = Ad$u_t ◦ α_t$, $t ∈ \boldsymbol{R}$ is also a flow on $\mathcal{B}$, where Ad$u_t(B) \triangleq u_tBu^{−1}_t$ for any $B ∈ \mathcal{B}$. $β$ is said to be a cocycle perturbation of $α$. We show that if $α$, $β$ are two flows on a nest algebra (or quasi-triangular algebra), then $β$ is a cocycle perturbation of $α$. And the flows on a nest algebra (or quasi-triangular algebra) are all uniformly continuous.     

Finite Gap Jacobi Matrices, II.?The Szeg? Class

Let \frake ì \mathbbR\frak{e}\subset\mathbb{R} be a finite union of disjoint closed intervals. We study measures whose essential support is \frake{\frak{e}} and whose discrete eigenvalues obey a 1/2-power condition. We show that a Szegő condition is equivalent to

Yangians and Mickelsson Algebras I

We study the composition of the functor from the category of modules over the Lie algebra to the category of modules over the degenerate affine Hecke algebra of GL_N introduced by I. Cherednik, with the functor from the latter category to the category of modules over the Yangian due to V. Drinfeld. We propose a representation theoretic explanation of a link between the intertwining operators on the tensor products of -modules, and the "extremal cocycle" on the Weyl group of defined by D. Zhelobenko. We also establish a connection between the composition of the functors, and the "centralizer construction" of the Yangian discovered by G. Olshanski.     

Density, Overcompleteness, and Localization of Frames. I. Theory

Frames have applications in numerous fields of mathematics and engineering. The fundamental property of frames which makes them so useful is their overcompleteness. In most applications, it is this overcompleteness that is exploited to yield a decomposition that is more stable, more robust, or more compact than is possible using nonredundant systems. This work presents a quantitative framework for describing the overcompleteness of frames. It introduces notions of localization and approximation between two frames and ( a discrete abelian group), relating the decay of the expansion of the elements of in terms of the elements of via a map . A fundamental set of equalities are shown between three seemingly unrelated quantities: The relative measure of , the relative measure of — both of which are determined by certain averages of inner products of frame elements with their corresponding dual frame elements — and the density of the set in . Fundamental new results are obtained on the excess and overcompleteness of frames, on the relationship between frame bounds and density, and on the structure of the dual frame of a localized frame. In a subsequent article, these results are applied to the case of Gabor frames, producing an array of new results as well as clarifying the meaning of existing results. The notion of localization and related approximation properties introduced in this article are a spectrum of ideas that quantify the degree to which elements of one frame can be approximated by elements of another frame. A comprehensive examination of the interrelations among these localization and approximation concepts is presented.     

Complex finitary simple Lie algebras

An algebra is called finitary if it consists of finite-rank transformations of a vector space. We classify finitary simple Lie algebras over an algebraically closed field of zero characteristic. It is shown that any such algebra is isomorphic to one of the following¶ (1) a special transvection algebra \frak t(V,P)\frak t(V,\mit\Pi );¶ (2) a finitary orthogonal algebra \frak fso (V,q)\frak {fso} (V,q); ¶ (3) a finitary symplectic algebra \frak fsp (V,s)\frak {fsp} (V,s).¶Here V is an infinite dimensional K-space; q (respectively, s) is a symmetric (respectively, skew-symmetric) nondegenerate bilinear form on V; and P\Pi is a subspace of the dual V^* whose annihilator in V is trivial: 0={v ? V | Pv=0}0=\{{v}\in V\mid \Pi {v}=0\}.     

Linearizing Systems of Second-Order ODEs via Symmetry Generators Spanning a Simple Subalgebra

It is shown that a system of n second order ordinary differential equations that possess 2(n?1) symmetries of certain type necessarily has maximal symmetry $\frak{sl}(n+2,\mathbb{R})$ . Further, it is shown for non-linearizable systems containing a subalgebra of symmetries isomorphic to $\frak{sl}(n-1,\mathbb{R})$ the dimension of the symmetry algebra $\mathcal{L}$ is d≥n ²?1. Examples showing that the upper bound is sharp are given.     

On $\frak {F}$-normal Fitting classes of finite soluble groups

Let \frak X, \frak F,\frak X\subseteqq \frak F\frak {X}, \frak {F},\frak {X}\subseteqq \frak {F}, be non-trivial Fitting classes of finite soluble groups such that G_{\frak X}G_{\frak {X}} is an \frak X\frak {X}-injector of G for all G ? \frak FG\in \frak {F}. Then \frak X\frak {X} is called \frak F\frak {F}-normal. If \frak F=\frak S_p\frak {F}=\frak {S}_{\pi }, it is known that (1) \frak X\frak {X} is \frak F\frak {F}-normal precisely when \frak X^*=\frak F^*\frak {X}^{\ast }=\frak {F}^{\ast }, and consequently (2) \frak F í \frak X\frak N\frak {F}\subseteq \frak {X}\frak {N} implies \frak X^*=\frak F^*\frak {X}^{\ast }=\frak {F}^{\ast }, and (3) there is a unique smallest \frak F\frak {F}-normal Fitting class. These assertions are not true in general. We show that there are Fitting classes \frak F\not = \frak S_p\frak {F}\not =\frak {S}_{\pi } filling property (1), whence the classes \frak S_p\frak {S}_{\pi } are not characterized by satisfying (1). Furthermore we prove that (2) holds true for all Fitting classes \frak F\frak {F} satisfying a certain extension property with respect to wreath products although there could be an \frak F\frak {F}-normal Fitting class outside the Lockett section of \frak F\frak {F}. Lastly, we show that for the important cases \frak F=\frak Nⁿ, n\geqq 2\frak {F}=\frak {N}^{n},\ n\geqq 2, and \frak F=\frak S_p1?\frak S_pr, p_i \frak {F}=\frak {S}_{p_{1}}\cdots \frak {S}_{p_{r}},\ p_{i} primes, there is a unique smallest \frak F\frak {F}-normal Fitting class, which we describe explicitly.