扭Heisenberg-Virasoro代数上Hom-李代数结构

Hom-李代数是一类满足反对称和Hom-Jacobi等式的非结合代数.扭Heisenberg-Virasoro代数是次数不超过1的微分算子代数的中心扩张,它是一类重要的无限维李代数,与一些曲线的模空间有关.文章主要研究扭Heisenberg-Virasoro代数上Hom-李代数结构,确定了扭Heisenberg-Virasoro代数上存在非平凡的Hom-李代数结构.     

可递Lie代数胚1阶上同调群的局部化

对连通流形M上可递Lie代数胚A的任意一个向量丛F上的表示, 研究一个称作局部化的同调群的同态¡ ^k: H^k (A, F)→H^k (L_x, F_x), 其中L_x是在x∈M处的伴随Lie代数. 主要结果是: 当底流形M单连通或者H⁰(L_x, F_x)平凡时, ¡¹是单射, 即Lie代数胚A的1阶上同调群由在点x∈M处伴随Lie代数 的1阶上同调群完全决定.     

量子环面李代数的自同构群, 泛中心扩张与导子

C_q:=C_q[x^±1₁, x^±1₂] 为复数域上的量子环面, 其中q≠ 0是一个非单位根, D(C_q) 为C_q的导子李代数. 记L_q 为C_q ㈩ D(C_q)的导出子代数. 该文研究李代数L_q的自同构群, 泛中心扩张和导子李代数.     

关于分次代数的一个Fong型定理

证明有限p-可解群G的特征p的域上的分次代数的任一投射不可分模是从它的一个Hall p′-子群H的分次子代数的模的诱导模; 并且给出了它的Hall p′-子群H的分次子代数的投射不可分模的诱导模仍然不可分的充分必要条件.     

扩张下表示的不变性质

讨论半单Hopf代数上经cleft扩张后的代数表示的不变性质. 首先,解释了cleft扩张的概念,并给出cleft扩张和交叉积之间的联系(交叉积是解决问题的基本方法). 然后,利用这些关系证明了: 当k是代数闭域, H是一个有限维的半单k代数时, 对一个有限维k代数, 其H-cleft扩张下的代数表示型是不变的. 另一方面证明了: 当k是任意域, H是一个有限维的半单k-Hopf代数,对一个根是H稳定的k代数,其Nakayama性质在H-cleft扩张下也是不变的.     

Hom-李代数的广义导子

给出Hom-李代数L的广义导子代数、拟导子代数和中心导子代数的一些基本性质.进一步地,有GDer(L)=QDer(L)+QC(L).同时,得到QDer(L)可以嵌入并成为一个更大的Hom-李代数的导子.     

广义Hom-李代数的中心不变量（英文）

设L是一个广义Hom-李代数,V是[L,L]的一个H-Hom-李理想.本文主要研究了L的中心不变量问题.利用Hopf代数中的方法,得到了V的H-不变量包含在L的中心H-不变量中,这推广了1994年Cohen和Westreich的主要结论.     

阶化Cartan型李代数的阶化模(Ⅱ)——正、负阶化模

本文对阶化李代数的正、负阶化模作了一般的讨论,所得的结果应用于阶化Cartan型李代数L的阶化模(?)V₀及(?)V₀。特别,得到了Tricomi算子表示及微分表示的内蕴解释,后者是L由以定义的。     

D4型量子包络代数的Gelfand-Kirillov维数

本文研究了D₄ 型量子包络代数的Gelfand-Kirillov 维数的计算问题. 利用文献[1] 中给出的Gelfand-Kirillov 维数的计算方法和文献[2] 中给出的D₄ 型量子包络代数的Groebner-Shirshov 基计算了D₄型量子包络代数的Gelfand-Kirillov 维数, 得到的主要结果是D₄ 型量子包络代数的Gelfand-Kirillov 维数为28. 希望此结果为计算D_n型量子包络代数的Gelfand-Kirillov 维数提供一些思路.     

Cartan型模李超代数S(m,n;t)的中心扩张

本文研究了特征为素数p>2的有限维Special李超代数S(m,n;t)的中心扩张.通过计算从S(m,n;t)到S(m,n;t)^*的斜外导子,得到二阶上同调群H²(S(m,n;t),F)是平凡的.应用此结果,可得S(m,n;t)的中心扩张是平凡的.     

A Frobenius–Schur Theorem for Hopf Algebras

We prove a version of the Frobenius–Schur theorem for a finite-dimensional semisimple Hopf algebra H over an algebraically closed field; if the field has characteristic p not 0, H is also assumed to be cosemisimple. Then for each irreducible representation V of H, we define a Schur indicator for V, which reduces to the classical Schur indicator when H is the group algebra of a finite group. We prove that this indicator is 0 if and only if V is not self-dual. If V is self dual, then the indicator is positive (respectively, negative) if and only if V admits a nondegenerate bilinear symmetric (resp., skew-symmetric) H-invariant form. A more general result is proved for algebras with involution.     

Holomorphic H-spherical distribution vectors in principal series representations

Let G/H be a semisimple symmetric space. The main tool to embed a principal series representation of G into L²(G/H) are the H-invariant distribution vectors. If G/H is a non-compactly causal symmetric space, then G/H can be realized as a boundary component of the complex crown . In this article we construct a minimal G-invariant subdomain _H of with G/H as Shilov boundary. Let be a spherical principal series representation of G. We show that the space of H-invariant distribution vectors of , which admit a holomorphic extension to _H, is one dimensional. Furthermore we give a spectral definition of a Hardy space corresponding to those distribution vectors. In particular we achieve a geometric realization of a multiplicity free subspace of L²(G/H)_mc in a space of holomorphic functions.     

Zur Idealtheorie in Gruppenalgebren von [FIA] B − -Gruppen

In this paper we consider closedB-invariant ideals in the group algebraL ¹(G), whereG is a locally compact group with a relatively compact groupB of topological automorphisms, which contains the set of all inner automorphisms. We study conditions when closedB-invariant ideals are completely determined by their hull. Also questions concerning the existence of approximate units in these ideals will be answered. Above all, we shall study these properties with regard to the relations between ideals inL ¹(G),L ¹ (G/N) andL ¹(N), whereN is a closedB-invariant subgroup ofG.     

Bi‐arc graphs and the complexity of list homomorphisms

Given graphs G, H, and lists L(v) ? V(H), v ε V(G), a list homomorphism of G to H with respect to the lists L is a mapping f : V(G) → V(H) such that uv ε E(G) implies f(u)f(v) ε E(H), and f(v) ε L(v) for all v ε V(G). The list homomorphism problem for a fixed graph H asks whether or not an input graph G, together with lists L(v) ? V(H), v ε V(G), admits a list homomorphism with respect to L. In two earlier papers, we classified the complexity of the list homomorphism problem in two important special cases: When H is a reflexive graph (every vertex has a loop), the problem is polynomial time solvable if H is an interval graph, and is NP‐complete otherwise. When H is an irreflexive graph (no vertex has a loop), the problem is polynomial time solvable if H is bipartite and H is a circular arc graph, and is NP‐complete otherwise. In this paper, we extend these classifications to arbitrary graphs H (each vertex may or may not have a loop). We introduce a new class of graphs, called bi‐arc graphs, which contains both reflexive interval graphs (and no other reflexive graphs), and bipartite graphs with circular arc complements (and no other irreflexive graphs). We show that the problem is polynomial time solvable when H is a bi‐arc graph, and is NP‐complete otherwise. In the case when H is a tree (with loops allowed), we give a simpler algorithm based on a structural characterization. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 61–80, 2003     

Functional analytic aspects of non-commutative harmonic analysis

Given a homogeneous space X = G/H with an invariant measure it is shown, using Grothendieck's inequality, that a G-invariant Hilbert subspace of the space of distributions of order zero on X is actually contained in L_loc²(X). Moreover, if θ is an automorphism on G appropriately related to H, it is shown that, under condition that H-orbits are smooth, an H-bi-invariant distribution of positive type on G satisfies the identity Ť^θ = T if the corresponding Hilbert space is contained in L_loc²(X). This shows that, under the smooth orbit condition, G-invariant Hilbert subspaces of L_loc² (X) have a unique decomposition into irreducible Hilbert spaces as in the case of generalized Gelfand pairs.     

Exponential actions and maximal ?-invariant ideals

Let V be an exponential ?-module, ? being an exponential Lie algebra. Put ? = exp ?. Then every orbit of V under the action of ? admits a closed orbit in its closure. If G= exp ? is a nilpotent Lie group and ? an exponential algebra of derivations of ?, then ? = exp ? acts on G, L ¹(G), (?) and the maximal ?-invariant ideals of L ¹(G), resp. of (?) coincide with the kernels Ker Ω, resp. Ker Ω∩ (?), where Ω is a closed orbit of ?^*. Received: 6 December 1996 / Revised version: 7 December 1997     

A reciprocity theorem for unitary representations of Lie groups

LetG be a Lie group,H a closed subgroup,L a unitary representation ofH andU ^L the corresponding induced representation onG. The main result of this paper, extending Ol’ŝanskii’s version of the Frobenius reciprocity theorem, expresses the intertwining number ofU ^L and an irreducible unitary representationV ofG in terms ofL and the restriction ofV _∞ toH.     

Isomorphisms preserving invariants

Let V and W be finite dimensional real vector spaces and let G ì GL(V){G \subset {\rm GL}(V)} and H ì GL(W){H \subset {\rm GL}(W)} be finite subgroups. Assume for simplicity that the actions contain no reflections. Let Y and Z denote the real algebraic varieties corresponding to \mathbbR[V]^G{\mathbb{R}[V]^G} and \mathbbR[W]^H{\mathbb{R}[W]^H}, respectively. If V and W are quasi-isomorphic, i.e., if there is a linear isomorphism L : V → W such that L sends G-orbits to H-orbits and L ⁻¹ sends H-orbits to G-orbits, then L induces an isomorphism of Y and Z. Conversely, suppose that f : Y → Z is a germ of a diffeomorphism sending the origin of Y to the origin of Z. Then we show that V and W are quasi-isomorphic, This result is closely related to a theorem of Strub [8], for which we give a new proof. We also give a new proof of a result of Kriegl et al. [3] on lifting of biholomorphisms of quotient spaces.     

Complement preserving operators

An operatorTVV on a real inner product space is called complement preserving if, wheneverU is aT-invariant subspace ofV the orthogonal complementU is alsoT-invariant. In this note we obtain some results on such operators.     

Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Let (E,H,μ) be an abstract Wiener space and let D_V:=VD, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space . Given a bounded operator B on , coercive on the range , we consider the operators A:=V^*BV in H and in , as well as the realisations of the operators and in L^p(E,μ) and respectively, where 1<p<∞. Our main result asserts that the following four assertions are equivalent:

(1) with for ;

(2) admits a bounded H^∞-functional calculus on ;

(3) with for ;

(4) admits a bounded H^∞-functional calculus on .

Moreover, if these conditions are satisfied, then . The equivalence (1)–(4) is a non-symmetric generalisation of the classical Meyer inequalities of Malliavin calculus (where , V=I, ). A one-sided version of (1)–(4), giving L^p-boundedness of the Riesz transform in terms of a square function estimate, is also obtained. As an application let −A generate an analytic C₀-contraction semigroup on a Hilbert space H and let −L be the L^p-realisation of the generator of its second quantisation. Our results imply that two-sided bounds for the Riesz transform of L are equivalent with the Kato square root property for A. The boundedness of the Riesz transform is used to obtain an L^p-domain characterisation for the operator L.