李Poisson超代数的泛中心扩张

研究了李Poisson超代数的泛中心扩张问题.通过构造其泛中心扩张,得到了其存在泛覆盖的充要条件是李Poisson超代数是完全的,并对李Poisson超代数的自同构群及导子的提升给出了结果.     

Universal Central Extensions of Lie Groups

We call a central Z-extension of a group G weakly universal for an Abelian group A if the correspondence assigning to a homomorphism ZA the corresponding A-extension yields a bijection of extension classes. The main problem discussed in this paper is the existence of central Lie group extensions of a connected Lie group G which is weakly universal for all Abelian Lie groups whose identity components are quotients of vector spaces by discrete subgroups. We call these Abelian groups regular. In the first part of the paper we deal with the corresponding question in the context of topological, Fréchet, and Banach–Lie algebras, and in the second part we turn to the groups. Here we start with a discussion of the weak universality for discrete Abelian groups and then turn to regular Lie groups A. The main results are a Recognition and a Characterization Theorem for weakly universal central extensions.     

Central Extensions of Steinberg Lie Superalgebras of Small Rank

It was shown by Mikhalev and Pinchuk (2000 Mikhalev , A. V. , Pinchuk , I. A. ( 2000 ). Universal central extensions of the matrix Lie superalgebras sl(m,n,A) . Int. Conf. in H.K.U., AMS , 111 – 125 . [Google Scholar]) that the second homology group H ₂(𝔰𝔱(m,n,R)) of the Steinberg Lie superalgebra 𝔰𝔱(m,n,R) is trivial for m + n ≥ 5. In this article, we will work out H ₂(𝔰𝔱(m,n,R)) explicitly for m + n = 3, 4.     

泛阶化李超代数

对于给定的负阶化李超代数K-,本文定义了K-型泛阶化李超代数并证明了它的存在性．进而引出阶化Cartan型李超代数,并且证得阶化Cartan型李超代数 W(m,n),K(m,n,ωA),S(m,n)和H(m,n)分别可以用某种泛阶化李超代数来刻画．     

δ-Jordan李超代数的交换扩张

通过δ-Jordan李超代数T的表示和上同调理论,构造δ-Jordan李超代数T■V.证明了δ-Jordan李超代数的等价交换扩张给出相同的表示.通过δ-Jordan李超代数的表示和其交换扩张得到2-上圈.     

李超代数的泛包络代数同构问题

李超代数的一个性质P叫做关于泛包络代数的不变量,如果对于任意李超代数L,H,只要L具有性质P,并且泛包络代数U(L)和U(H)作为结合超代数是同构的,那么H亦具有性质P.通过讨论李超代数关于泛包络代数的不变量证明了:如果L的幂零长度不超过2,那么L和H是同构的.     

Universal Central Extensions in Semi-Abelian Categories

Basing ourselves on Janelidze and Kelly’s general notion of central extension, we study universal central extensions in the context of semi-abelian categories. We consider a new fundamental condition on composition of central extensions and give examples of categories which do, or do not, satisfy this condition.     

On Universal Central Extensions of Leibniz Algebras

We construct the endofunctor 𝔲𝔠𝔢 between the category of Leibniz algebras which assigns to a perfect Leibniz algebra its universal central extension, and we obtain the isomorphism 𝔲𝔠𝔢_Lie(𝔮_Lie) ? (𝔲𝔠𝔢_Leib(𝔮))_Lie, where 𝔮 is a perfect Leibniz algebra satisfying the condition [x, [x, y]] + [[x, y], x] = 0, for all x, y ∈ 𝔮. Moreover, we obtain several results concerning the lifting of automorphisms and derivations in a covering. We also study the relationship between the universal central extension of a semidirect product of perfect Leibniz algebras and the semidirect product of the universal central extension of both of them.     

完备Lie超代数

本文引入了完备Ｌｉｅ超代数和Ｌｉｅ超代数的全形这两个概念，讨论了完备Ｌｉｅ超代数的一些等价条件和结构定理。所得结果是Ｊａｃｏｂｓｏ［１］和ＮｅｎｇＤａｏｊｉ［２］的推广。     

Central Extensions of Matrix Lie Superalgebras Over \mathbb{Z}/2\mathbb{Z}-graded Algebras

We study central extensions of the Lie superalgebra $\mathfrak{s}\mathfrak{l}_n(A)$ when A is a ?/2?-graded superalgebra over a commutative ring K. Steinberg Lie superalgebras and their central extensions play an essential role. We use a ?/2?-graded version of cyclic homology to study the center of the extensions in question.     

Leibniz Central Extensions on Some Infinite-Dimensional Lie Algebras

Abstract

In this paper, all one-dimensional Leibniz central extensions on the algebras of differential operators over C[t, t ^?1] and C((t)), as well as on the quantum 2-torus, the Virasoro-like algebra and its q-analog are studied. We determine all nontrivial Leibniz 2-cocycles on these infinite dimensional Lie algebras.     

李超代数拟型心

定义了李超代数的拟型心和型心.利用拟型心的Fitting分解研究了拟型心的结构.     

关于完满的Lie超代数

In this paper, some properties of perfect Lie superalgebras are investigated. We prove that the derivation superalgebra of a centerless perfect Lie superalgebra of arbitrary dimension over a field of arbitrary characteristic is complete and we obtain a necessary and sufficient condition for the holomorph of a centerless perfect Lie superalgebra to be complete. Finally, some properties of perfect restricted Lie superalgebras are given.     

Unimodality and Lie Superalgebras

It is well-known how the representation theory of the Lie algebra sl(2, ?) can be used to prove that certain sequences of integers are unimodal and that certain posets have the Sperner property. Here an analogous theory is developed for the Lie superalgebra osp(1,2). We obtain new classes of unimodal sequences (described in terms of cycle index polynomials) and a new class of posets (the “super analogue” of the lattice L(m,n) of Young diagrams contained in an m × n rectangle) which have the Sperner property.     

Super-biderivations and Post-Lie Superalgebras on Some Lie Superalgebras

In this paper,all symmetric super-biderivations of some Lie superalgebras are determined.As an application,commutative post-Lie super algebra structures on these Lie super algebras are also obtained.     

Modular Lie Superalgebras of H-type

In this article the Z-graded transitive modular Lie superalgebra -1 Li,whose repre-sentation of L_0 in L_(-1) is isomorphic to the natural representation of osp(L_(-1)),is determined.     

Representation Rings of Lie Superalgebras

Given a Lie superalgebra , we introduce several variants of the representation ring, built as subrings and quotients of the ring of virtual -supermodules, up to (even) isomorphisms. In particular, we consider the ideal of virtual -supermodules isomorphic to their own parity reversals, as well as an equivariant K-theoretic super representation ring on which the parity reversal operator takes the class of a virtual -supermodule to its negative. We also construct representation groups built from ungraded -modules, as well as degree-shifted representation groups using Clifford modules. The full super representation ring , including all degree shifts, is then a -graded ring in the complex case and a -graded ring in the real case. Our primary result is a six-term periodic exact sequence relating the rings , and . We first establish a version of it working over an arbitrary (not necessarily algebraically closed) field of characteristic 0. In the complex case, this six-term periodic long exact sequence splits into two three-term sequences, which gives us additional insight into the structure of the complex super representation ring . In the real case, we obtain the expected 24-term version, as well as a surprising six-term version, of this periodic exact sequence. (Received: October 2004)     

Modular Lie Superalgebras of H-type

In this article the Z-graded transitive modular Lie superalgebra О i≥-1 Li, whose representation of Lo in L-1 is isomorphic to the natural representation of osp(L-1), is determined.     

Calogero Operator and Lie Superalgebras

We construct a supersymmetric analogue of the Calogero operator , which depends on the parameter k. This analogue is related to the root system of the Lie superalgebra . It becomes the standard Calogero operator for m = 0 and becomes the operator constructed by Veselov, Chalykh, and Feigin up to changing the variables and the parameter k for m = 1. For k = 1 and 1/2, the operator is the radial part of the second-order Laplace operator for the symmetric superspaces corresponding to the respective pairs . We show that for any m and n, the supersymmetric analogues of the Jack polynomials constructed by Kerov, Okounkov, and Olshanskii are eigenfunctions of the operator . For k = 1 and 1/2, the supersymmetric analogues of the Jack polynomials coincide with the spherical functions on the above superspaces. We also study the algebraic analogue of the Berezin integral.