有限维非退化可解李代数的顶点算子代数

构造相应于非退化可解李代数g的顶点算子代数分两步进行,首先构造顶点代数.本文是在已经得到的相应于非退化可解李代数g的顶点代数(Vg(l,0),Y(V,1)上构造顶点算子代数.定义了非退化可解李代数g的Casimir算子Ω,给出了在伴随表示下Ω作用在g上是0及相关性质,并应用Ω定义出Vg(l,0)中元素ω,证明了Vg(l,0)关于ω的顶点算子YV(ω,x)的系数构成一个Virasoro代数-模,还证明了ω满足顶点算子代数定义中Virasoro-向量的所有公理.从而证得(Vg(l,0),Yv,1,ω)是一个顶点算子代数.     

构造相应于有限维非退化可解李代数的顶点代数

设g是带有非退化不变对称双线性型的有限维可解李代数, 该文首先应用g的仿射李代数{\heiti $\hat{g}$}的表示理论,构造出一类水平为l的限制$\hat{g}$ -模$V_{\hat{g}}(l,0)$.然后应用顶点算子的局部理论在hom$(V_{\hat{g}}(l,0),V_{\hat{g}}(l,0)((x)))$中 找到一类顶点代数$L_{V_{\hat{g}}(l,0)}$.建立了$L_{V_{\hat{g}}(l,0)}$到 $V_{\hat{g}}(l,0)$的映射,最后证明了这类映射是顶点代数同构.     

顶点代数V_(l,0)的极小生成问题

设g是有限维单李代数,是相应于g的无扭仿型Kac-Moody代数的导代数.讨论了相应于的顶点代数V_(l,0)的极小生成问题,证明了V_(l,0)作为顶点代数由a,h两个元素生成,其中a,h∈g.     

一般扭算子李代数

由扭算子构成的扭算子李代数在李代数理论中占有重要的位置,首先构造了一般形式的扭顶点算子Z~σ(E_(ij),α,β,z),然后给出了一般扭算子李代数g(G,l)[σ],研究了一般扭顶点算子所具有的性质.     

广义Baby-TKK李代数的一类顶点表示

利用广义 Virasoro- Toroidal李代数的顶点表示理论研究了广义 Baby- TKK李代数的一类顶点表示 .     

关于局部顶点李代数的一些注记

本文研究局部顶点李代数与顶点代数之间的关系.利用由局部顶点李代数构造顶点代数的方法,定义局部顶点李代数之间的同态,证明了同态可以唯一诱导出由局部顶点李代数构造所得到的顶点代数之间的同态.     

Tits-Kantor-Koecher李代数的Wakimoto表示

每一个Jordan代数都对应了一个Tits-Kantor-Koecher李代数．在扩张仿射李代数的分类中[1],A_1型李代数的分类依赖于欧氏空间上半格给出的Tits-Kantor-Koecher李代数．另外在相似的意义下,二维欧氏空间R~2中只有两个半格．设S是R~2上的任一半格,T(S)为半格S对应的Jordan代数,G(T(S))为相应的Tits-Kantor-Koecher李代数．利用Wakimoto自由场的方法给出李代数G(T(S))的一类顶点表示．     

Tits-Kantor-Koecher李代数的Wakimoto表示

每一个Jordan代数都对应了一个Tits-Kantor-Koecher李代数.在扩张仿射李代数的分类中[1],A1型李代数的分类依赖于欧氏空间上半格给出的Tits-Kantor-Koecher李代数.另外在相似的意义下,二维欧氏空间R2中只有两个半格.设S是R2上的任一半格,Τ(S)为半格S对应的Jordan代数,(g)(Τ(S))为相应的Tits.Kantor-Koecher李代数.利用Wakimoto自由场的方法给出李代数(g)(Τ(S))的一类顶点表示.     

广义Virasoro-Toroidal李代数不可约模

本文将Kac-Moody代数A1（1）的二阶表示理论[11]推广到Toroidal李代数的情形.并给出了A1型Toroidal李代数的一类不可约表示.     

一般扭算子李代数g(G,l)[σ]的结构

扭算子李代数在研究李代数的结构中有着广泛的应用,因而讨论扭算子李代数的结构具有很重要的意义.讨论了随着G,l的选取,在各种情形下扭算子李代数g(G,l)[σ]所具有的代数结构.     

有限秩算子的表示（I）

定义了子空间格代数的（弱闭双边）模,对有限维Hilbert空间的强自反子空间格代数的模及原子Boolean格代数的模中的有限秩算子进行了讨论,得到了有限秩算子一定可以表示为秩1算子的和。     

Dixmier's Problem 6 for the Weyl Algebra (The Generic Type Problem)

ABSTRACT

In Dixmier (1968 Dixmier , J. ( 1968 ). Sur les algèbres de Weyl . Bull. Soc. Math. France 96 : 209 – 242 . [CSA] [Crossref] , [Google Scholar]), the author posed six problems for the Weyl algebra A ₁ over a field K of characteristic zero. Problems 3, 6, and 5 were solved respectively by Joseph (1975 Joseph , A. ( 1975 ). The Weyl algebra—semisimple and nilpotent elements . Amer. J. Math. 97 ( 3 ): 597 – 615 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) and Bavula (2005a Bavula , V. V. ( 2005a ). Dixmier's Problem 5 for the Weyl algebra . J. Algebra 283 ( 2 ): 604 – 621 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]). Problems 1, 2, and 4 are still open. In this article a short proof is given to Dixmier's problem 6 for the ring of differential operators 𝒟 (X) on a smooth irreducible algebraic curve X. It is proven that, for a given maximal commutative subalgebra C of 𝒟 (X), (almost) all noncentral elements of it have the same type, more precisely, have exactly one of the following types: (i) strongly nilpotent; (ii) weakly nilpotent; (iii) generic; (iv) generic, except for a subset K*a + K of strongly semi-simple elements; (iv) generic, except for a subset K*a + K of weakly semi-simple elements, where K* := K\{0}. The same results are true for other popular algebras.     

Automorphisms of the algebra ofq-difference operators

In this paper, automorphisms of the algebra ofq-difference operators, as an associative algebra for arbitraryq and as a Lie algebra forq being not a root of unity, are determined. Project supported by the NNSF of China     

Decomposition of

For any locally compact group , let and be the Fourier and the Fourier-Stieltjes algebras of , respectively. is decomposed as a direct sum of and , where is a subspace of consisting of all elements that satisfy the property: for any and any compact subset , there is an with and such that is characterized by the following: an element is in if and only if, for any there is a compact subset such that for all with and . Note that we do not assume the amenability of . Consequently, we have for all if is noncompact. We will apply this characterization of to investigate the general properties of and we will see that is not a subalgebra of even for abelian locally compact groups. If is an amenable locally compact group, then is the subspace of consisting of all elements with the property that for any compact subset , .

     

Indecomposable modules of the intermediate series over W(a, b)

For any complex parameters a and b,W(a,b)is the Lie algebra with basis{Li,Wi|i∈Z}and relations[Li,Lj]=(j i)Li+j,[Li,Wj]=(a+j+bi)Wi+j,[Wi,Wj]=0.In this paper,indecomposable modules of the intermediate series over W(a,b)are classified.It is also proved that an irreducible Harish-Chandra W(a,b)-module is either a highest/lowest weight module or a uniformly bounded module.Furthermore,if a∈/Q,an irreducible weight W(a,b)-module is simply a Vir-module with trivial actions of Wk’s.     

Second Homology Groups and Universal Coverings of Steinberg Leibniz Algebras of Small Characteristic

It is known that the second Leibniz homology group HL ₂ (𝔰𝔱𝔩 _n (R)) of the Steinberg Leibniz algebra 𝔰𝔱𝔩 _n (R) is trivial for n ≥ 5. In this article, we determine HL ₂(𝔰𝔱𝔩 _n (R)) explicitly (which are shown to be not necessarily trivial) for n = 3, 4 without any assumption on the base ring.     

Norm Estimates for Multiplication Operators in Hilbert Algebras

In this paper, it is proved that for the bilinear operator defined by the operation of multiplication in an arbitrary associative algebra with unit over the fields or , the infimum of its norms with respect to all scalar products in this algebra (with ) is either infinite or at most . Sufficient conditions for this bound to be not less than are obtained. The finiteness of this bound for infinite-dimensional Grassmann algebras was first proved by Kupsh and Smolyanov (this was used for constructing a functional representation for Fock superalgebras).     

Composition algebras of the second kind

The concept of a composition algebra of the second kind is introduced. We prove that such algebras are non-degenerate monocomposition algebras without unity. A big number of these algebras in any finite dimension are constructed, as well as two algebras in a countable dimension. The constructed algebras each contains a non-isotropic idempotent e² = e. We describe all orthogonally non-isomorphic composition algebras of the second kind in the following forms: (1) a two-dimensional algebra (which has turned out to be unique); (2) three-dimensional algebras in the constructed series. For every algebra A, the group Ortaut A of orthogonal automorphisms is specified. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 428–447, July–August, 2007.