李超代数gl_(2|1)到Witt超代数的低维上同调

Witt超代数在伴随表示的意义下是一般线性李超代数gl_(2|1^-)模.通过对Witt超代数进行子模分解和权空间分解,用简约的方法计算了gl_(2|1-)模.通过对Witt超代数进行子模分解和权空间分解,用简约的方法计算了gl_(2|1^-)到Witt超代数的低维上同调.     

q-Schur超代数

本文总结了近期在q-Schur超代数、量子一般线性超群和它们的典范基以及不可约(多项式)表示方面的研究.首先给出了q-Schur超代数在三种不同背景下的定义和相应的基,并且刻画了这三组基之间的关系,接着描述了q-Schur超代数中的某些乘法公式及其在量子一般线性超群的新实现、q-Schur超代数的正则表示和量子一般线性超群的正部分的典范基的构造中的应用,同时给出了q-Schur超代数的半单性的判别条件.通过对Alperin的权猜想和Scott的置换表示理论的推广,本文得到了q-Schur超代数的不可约模分类.本文最后提到了在不引入量子坐标代数情形下构造无穷小和小q-Schur超代数的新方法.     

一类深度1的可迁模Lie超代数

建立了满足如下条件的可迁$\mathbb{Z}$-分次模Lie超代数$\frak{g}=\oplus_{-1\leq i\leq r}\frak{g}_{i}$的嵌入定理:(i) $\frak{g}_{0}\simeq \widetilde{\mathrm{p}}(\frak{g}_{-1}) $ 并且$\frak{g}_{0}$-模 $\frak{g}_{-1}$ 同构于$\widetilde{\mathrm{p}}(\frak{g}_{-1})$的自然模;(ii) $\dim \frak{g}_1=\frac 23 n(2n^2+1),$ 其中 $n=\frac{1}{2} \dim \frak{g}_{-1}.$特别地, 证明了满足上述条件的有限维单模Lie超代数同构于奇Hamilton模Lie超代数.对局限Lie超代数也做了相应的讨论.     

套子代数上线性映射的Lie不变子空间

本文引入了Banach代数上线性映射的Lie不变子空间,给出了因子VonNeumann代数中套子代数上以导子空间为Lie不变子空间的线性映射的一般形式,研究了Lie导子与Lie自同构的概念及了Lie导子与Lie自同构半群的关系.     

psl(∧(2|2))(2)k 非线性σ-模型超对称性质研究

由psl(∧(2|2))(2)k非线性σ-模型加上WZ-项得到的WZW模型是共形场论,它具有李超代数psi(2|2)对称性.该文用向量相干态方法给出了李超代数psl(2|2)的微分算子表示.并在此基础上给出了扭曲Kac-Moody李超代数psl(∧(2|2))(2)k自由场实现,相应共形场论的中心荷为-2.     

实Lie超代数的有限维实不可约表示

<正> 本文主要是将有限维实Lie超代数的有限维实不可约表示的分类归结为有限维复Lie超代数的有限维复不可约表示的分类问题(为了简单,以下将“有限维”字样统统略去).由于复的可解与单纯Lie超代数的不可约表示是已知的,从而复的可解与单纯     

广义耦合方程的延拓结构

该文利用延拓结构理论及单(半单)Lie代数的性质,研究了两组对偶系统的延拓结构.并且利用Lie代数表示理论,给出了两组对偶系统的Lax对表示.对于Camassa-Holm(CH)型的方程,通过对其超定方程的分析,仅仅选择了阶数小于等于2的函数F进行讨论,然而经过计算与分析却只存在阶数为1的情况.     

平面向量及其运算

1　重、难点分析本单元学习的重点是 :1)向量的概念 ;2 )向量的运算及其性质 ;3)向量及其运算的坐标表示 .我们知道 ,在平面上取定一点O后 ,平面上的任意点P就与向量OP成一一对应 ,这样关于点的几何问题就与向量联系起来 ,由于向量可以进行运算 ,因此通过向量也就把代数运算引入到几何中 .所以 ,用代数的方法 (向量运算的方法 )处理几何问题是本单元内容中渗透的重要数学思想方法 .具体地 ,由向量的线性运算 (向量的加法、实数与向量的积 )可以得到两向量平行的充要条件及定比分点公式 ;由向量的数量积运算可以得到两向量垂直的充要条件及…     

Block型Lie超代数上的非权单模

本文研究Block型Lie超代数S(q)上的一类非权模.本文构造和分类了Ramond-Block代数上秩为1的U(η)-自由模和Neveu-Schwarz-Block代数上秩为2的U(h)-自由模.同时也给出上述两类模成为单模的充要条件,并决定了它们的同构类.本文中的结果包含了一些已有结果.     

psl(2|2) 非线性α -模型超对称性质研究

由$\widehat{psl(2|2)^{(2)}}_{k}$非线性$\sigma$ -模型加上WZ -项得到的WZW模型是共形场论,它具有李超代数$psl(2|2)$对称性.该文用向量相干态方法给出了李超代数$psl(2|2)$的微分算子表示.并在此基础上给出了扭曲Kac-Moody李超代数 $\widehat{psl(2|2)^{(2)}}_{k}$自由场实现,相应共形场论的中心荷为$-2$.     

Restricted Kac modules for special contact Lie superalgebras of odd type

We consider the simple restricted modules for special contact Lie superalgebras of odd type over an algebraically closed field of characteristic p>3: We give a suffcient and necessary condition in terms of typical or atypical weights for restricted Kac modules to be simple. In the process, we also determine the socle for each restricted Kac module and the length for each simple restricted module.     

Composition factors of the kac module for the lie superalgebras sl(3/3) and sl(r/r)

The central results of this article are two theorems on G _ō and G primitivity on relative level 1 of the Kac module ?(?) for the Lie superal-gebra G = sl(m/n). These theorems are combined with Kac module homomorphisms in order to prove a conjecture by Hughes et al. [HKV] on the number of composition factors of the Kac module for 3- and r- fold atypical highest weights of sl(3/3) and sl(rZ/r) respectively. Our theorems on G _ō and G primitivity are general in nature and therefore the results also include infinite-dimensional representations.     

On the Feynman–Kac representation for solutions of the Cauchy problem for parabolic equations in divergence form

We consider the Cauchy problem for general second–order uniformly elliptic linear equation in divergence form. We give a stochastic representation of bounded weak solutions of the problem in terms of solutions of associated linear backward stochastic differential equations. Our representation may be considered as an extension of the classical Feynman–Kac formula.     

Representations of the Witt Lie Superalgebra with <Emphasis Type="Italic">p</Emphasis>-character of Height One

Let W(n) be the Witt Lie superalgebra over an algebraically closed field k of characteristic p>2. We study the representations of W(n),n≥3 with p-character of height one. We prove that the Kac module always has a unique simple quotient and all simple modules are given in this way. In fact, most Kac modules are simple. We will determine the sufficient and necessary conditions such that the Kac module is simple. Moreover, composition factors of each Kac module can be determined. It is proved that the number of composition factors in each Kac module is at most 2.     

A recursive pricing formula for a path-dependent option under the constant elasticity of variance diffusion

In this paper, we consider a path-dependent option in finance under the constant elasticity of variance diffusion. We use a perturbation argument and the probabilistic representation (the Feynman–Kac theorem) of a partial differential equation to obtain a complete asymptotic expansion of the option price in a recursive manner based on the Black–Scholes formula and prove rigorously the existence of the expansion with a convergence error.     

Backward problems for stochastic differential equations on the Sierpinski gasket

In this paper, we study the non-linear backward problems (with deterministic or stochastic durations) of stochastic differential equations on the Sierpinski gasket. We prove the existence and uniqueness of solutions of backward stochastic differential equations driven by Brownian martingale (defined in Section 2) on the Sierpinski gasket constructed by S. Goldstein and S. Kusuoka. The exponential integrability of quadratic processes for martingale additive functionals is obtained, and as an application, a Feynman–Kac representation formula for weak solutions of semi-linear parabolic PDEs on the gasket is also established.     

Stochastic Burgers' equation on the real line: regularity and moment estimates

In this project, we investigate the stochastic Burgers' equation with multiplicative space-time white noise on an unbounded spatial domain. We give a random field solution to this equation by defining a process via a kind of Feynman–Kac representation which solves a stochastic partial differential equation such that its Hopf–Cole transformation solves Burgers' equation. Finally, we obtain Hölder regularity and moment estimates for the solution to Burgers' equation.     

Reflected BSDEs with nonpositive jumps,and controller-and-stopper games

We study a class of reflected backward stochastic differential equations with nonpositive jumps and upper barrier. Existence and uniqueness of a minimal solution are proved by a double penalization approach under regularity assumptions on the obstacle. In a suitable regime switching diffusion framework, we show the connection between our class of BSDEs and fully nonlinear variational inequalities. Our BSDE representation provides in particular a Feynman–Kac type formula for PDEs associated to general zero-sum stochastic differential controller-and-stopper games, where control affects both drift and diffusion term, and the diffusion coefficient can be degenerate. Moreover, we state a dual game formula of this BSDE minimal solution involving equivalent change of probability measures, and discount processes. This gives in particular a new representation for zero-sum stochastic differential controller-and-stopper games.     

Kazhdan-Lusztig polynomials and character formula for the Lie superalgebragI(m/n)

We find the character formula for irreducible finite-dimensional -modules. Also multiplicities of the composition factors in a Kac module are calculated.     

Classification of harish-chandra modules over the super-virasoro algebras

In this paper, we first construct all indecomposable modules whose dimensions of weight spaces of the even and odd parts are ≤ 1, then classify all Harish-Chandra module over the super-Virasoro algebras, proving that every Harish-Chandra module over the super-Virasoro algebras is either a highest or lowest weight module, or else a module of the intermediate series. This result generalizes a theorem which was originally given as a conjecture by Kac on the Virasoro algebra.