Cn中单位球上$mu$-Bloch函数的等价刻画

设$\mu$是$[0,1)$上的正规函数, 给出了${\bf C}^{\it n}$中单位球$B$上$\mu$-Bloch空间$\beta_{\mu}$中函数的几种刻画. 证明了下列条件是等价的: (1) $f\in \beta_{\mu}$; \ (2) $f\in H(B)$且函数$\mu(|z|)(1-|z|^{2})^{\gamma-1}R^{\alpha,\gamma}f(z)$ 在$B$上有界; (3) $f\in H(B)$ 且函数${\mu(|z|)(1-|z|^{2})^{M_{1}-1}\frac{\partial^{M_{1}} f}{\partial z^{m}}(z)}$ 在$B$上有界, 其中$|m|=M_{1}$; (4) $f\in H(B)$ 且函数${\mu(|z|)(1-|z|^{2})^{M_{2}-1}R^{(M_{2})}f(z)}$ 在$B$上有界.     

辫子Hopf代数的积分(英)

本文引进了无限维辫子Hopf代数$H$的忠实拟对偶$H^d$和严格拟对偶$H^{d'}$.证明了每个严格拟对偶$H^{d'}$是一个$H$-Hopf 模. 发现了$H^{d}$的极大有理$H^{d}$-子模$H^{d {\rm rat} }$ 与积分的关系, 即: $H^{d {\rm rat}}\cong \int ^l_{H^d} \otimes H$.给出了在Yetter-Drinfeld范畴$(^B_B{\cal YD},C)$中的辫子Hopf代数的积分的存在性和唯一性.     

$M_{2}({\mathbb F})$上的$H$-模代数结构

设${\mathbb F}$是特征为零的代数闭域, $H$为非点化非幺模的8维非半单Hopf代数, $M_{2}({\mathbb F})$为${\mathbb F}$上二阶方阵组成的全矩阵代数. 本文的主要目的是讨论和分类$M_{2}({\mathbb F})$上所有的$H$-模代数结构.     

Bloch型空间上一个积分算子的有界性

本文研究了单位圆盘$\mathbb{D}$上的一个积分算子,定义如下$Kf(z)=\int_{\mathbb{D}}\frac{f(w)}{1-z\bar{w}}{\rm d}A(w)$, 该算子可以看作经典Bergman投影的姐妹算子,同时我们得到了该算子关于Bloch型空间, $H^{\infty}$空间以及$L^{p}$空间之间有界的充分必要条件.     

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.     

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.     

由H\"{o}rmander向量场构成的抛物方程的 $W_{\ast}^{1,p}$ 正则性

设 $X_{1},\cdots ,X_{q}\ (q相似文献   

Heisenberg群上与Schr\"odinger算子相关的Poisson半群的分数阶导数估计

令$\mathcal{L}=-{\Delta}_{\mathbb{H}^{n}}+V$为Heisenberg群$\mathbb{H}^{n}$上的Schr\"odinger算子, 其中${\Delta}_{\mathbb{H}^{n}}$为次Laplace算子, 非负位势$V$属于逆H\"{o}lder类. 本文中, 利用从属性公式, 我们给出与$\mathcal{L}$相关的Poisson半群的分数阶导数的正则性估计, 作为应用, 我们得到了与$\mathcal{L}$相关的Campanato型空间的一个刻画.     

从$L^{\infty}$空间到Bloch型空间一类奇异积分算子的范数

本文研究了单位圆盘上从$L^{\infty}(\mathbb{D})$空间到Bloch型空间 $\mathcal{B}_\alpha$ 一类奇异积分算子$Q_\alpha, \alpha>0$的范数, 该算子可以看成投影算子$P$ 的推广,定义如下$$Q_\alpha f(z)=\alpha \int_{\mathbb{D}}\frac{f(w)}{(1-z\bar{w})^{\alpha+1}}\d A(w),$$ 同时我们也得到了该算子从 $C(\overline{\mathbb{D}})$空间到小Bloch型空间$\mathcal{B}_{\alpha,0}$上的范数.     

A FINITE STRUCTURE THEOREM BETWEEN PRIMITIVE RINGS AND ITS APPLICATION TO GALOIS THEORY

设\[\mathfrak{M} = \sum {F{u_i}} \]是除环F上向量空间，P是F的一个子除环且在F中是Galois，即存 在F的一个自同构群G使\[I(G) = P\].记Ф是F的中心，\[{G_0}\]是属于G的内自同构群， \[{G_0}\]的元素记为\[{I_r},r \in F\]；，记\[{E^'} = \sum\limits_{{I_{{r_j}}} \in {G_0}} {{\Phi _{{r_j}}}} \]是G的代数，\[P' = {C_F}({E^'})\]是\[{E^'}\]在F中的中心化子.记\[\mathfrak{U}(F,\mathfrak{M})\]是\[\mathfrak{M}\]的F-线性变换完全环，\[{T_v}(F,\mathfrak{M})\]是\[\mathfrak{U}(F,\mathfrak{M})\]中所有秩小于\[\mathcal{X}{_v}\]的元素集合，那末我们有如下主要结果： (1)\[{[F:P']_L} = n\]有限当且仅当\[{T_v}(P',\mathfrak{M}) = \sum\limits_{j = 1}^n \oplus {r_{jL}}{T_v}(F,\mathfrak{M})\]，其中\[{r_j} \in {E^'}\]，\[{r_{jL}}\]表示元素\[{r_j}\]的标量左乘. （2）\[{[P':P]_L} = t\]有限当且仅当凡\[{T_v}(P,\mathfrak{M}) = \sum\limits_{j = 1}^t \oplus {S_j}{T_v}(P',\mathfrak{M})\],其中\[{S_j}\]表示\[\mathfrak{M}\]的F-半线变换自同构，它的伴随同构\[{\psi _j} \in G\]. ⑶如有某个序数v使\[{T_v}(P,\mathfrak{M})\]，\[{T_v}(P',\mathfrak{M})\]及\[{T_v}(F,\mathfrak{M})\]满足⑴及(2)中的关系 式，那末对任何\[{T_\mu }(P,\mathfrak{M})\]，\[{T_\mu }(P',\mathfrak{M})\]及\[{T_\mu }(F,\mathfrak{M})\]皆满足(1)及(2)中的关系式.特别 对\[\mathfrak{U}(P,\mathfrak{M})\]，\[\mathfrak{U}(P',\mathfrak{M})\]及\[\mathfrak{U}(F,\mathfrak{M})\]也是如此. ⑷如果\[{[F:P]_L}\]有限，那末必有\[{C_p}({C_F}(E')) = E'\]，\[{[F:P']_L} = \dim E'\]，\[{[P':P']_L} = [G/{G_0}]\]，其中dim E'表示E'在\[\Phi \]上的维数，\[[G/{G_0}]\]表示\[{G_0}\]在G中的指数,特别\[G\]是 Galois 群，则 \[{C_F}(P') = {C_F}(P) = E'\]. (5)若\[{\tilde G}\]是F的另一自同构群且\[I(G) = I(\tilde G)\]，那末必有\[[G/{G_0}] = [\tilde G/{{\tilde G}_0}]\]， \[\dim {\kern 1pt} {\kern 1pt} E' = \dim {\kern 1pt} {\kern 1pt} \tilde E'\]. 其中\[{\kern 1pt} \tilde E'\]表示\[{\tilde G}\]的代数. 如果P取为F的中心时，于是从上述结果(1)就得出熟知的定理：\[[F:\Phi ]\]是有限的当 且仅当\[\mathfrak{U}(\Phi ,\mathfrak{M}) = \mathfrak{U}(F,\mathfrak{M}){ \otimes _\Phi }{F_L}\]. 另方面，运用我们上述的结果，可导出除环F的有限Galois理论.     

关于Witt代数基本子代数的一点注记

设g=W_1是特征p3的代数闭域k上的Witt代数.本文确定了g的极大基本子代数.进一步具体给出了最大维数的基本子代数的G共轭类,这里G是g的自同构群.从而证明了最大维数为(p-1)/2的基本子代数射影簇E((p-1)/2,g)是不可约的且是一维的.更进一步,证明了E(1,g)是不可约的,具有维数p-2,而E(2,g)是等维的,共有(p-3)/2个不可约分支,且每个不可约分支的维数是p-4.而当3≤r≤(p-3)/2时,E(r,g)是可约的.给出了E(r,g)(3≤r≤(p-3)/2)维数的一个下界.     

Schur–Weyl Theory for C*‐algebras

To each irreducible infinite dimensional representation $(\pi ,\mathcal {H})$ of a C*‐algebra $\mathcal {A}$, we associate a collection of irreducible norm‐continuous unitary representations $\pi _{\lambda }^\mathcal {A}$ of its unitary group ${\rm U}(\mathcal {A})$, whose equivalence classes are parameterized by highest weights in the same way as the irreducible bounded unitary representations of the group ${\rm U}_\infty (\mathcal {H}) = {\rm U}(\mathcal {H}) \cap (\mathbf {1} + K(\mathcal {H}))$ are. These are precisely the representations arising in the decomposition of the tensor products $\mathcal {H}^{\otimes n} \otimes (\mathcal {H}^*)^{\otimes m}$ under ${\rm U}(\mathcal {A})$. We show that these representations can be realized by sections of holomorphic line bundles over homogeneous Kähler manifolds on which ${\rm U}(\mathcal {A})$ acts transitively and that the corresponding norm‐closed momentum sets $I_{\pi _\lambda ^\mathcal {A}}^{\bf n} \subseteq {\mathfrak u}(\mathcal {A})^{\prime }$ distinguish inequivalent representations of this type.     

On the Springer''s Conjecture of the Coefficients of Univalent Meromorphic Functions

Let $\sigma$ denote the family of univalent functions $\[F(z) = z + \sum\limits_{n = 1}^\infty {\frac{{{b_n}}}{{{z^n}}}} \]$ in l< |z| <\infty if G(w) is the inverse of a function $F(z) \in \sigma ^'$, the expansion of G(w) in some neighborhood of w=\infty is $\[G(w) = w - \sum\limits_{n = 1}^\infty {\frac{{{B_n}}}{{{w^n}}}} \]$ It is well known that |B_1|\leq 1 for any F(z) \in \sigma ^'. Springer^[1] proved that | B_3| \leq 1 and conjectured that $\[|{B_{2n - 1}}| \le \frac{{(2n - 2)!}}{{n!(n - 1)!}}{\rm{ }}(n = 3,4, \cdots )\]$ (1) Kubota^[2] proved (1) for n=3, 4, 5. Schober^[3] proved (1) for n = 6, 7. Ren Fuyao[4,5] has verified (1) for n=6, 7, 8. In this article we are going to verify (1) for n=9.     

On the Kashiwara–Vergne conjecture

Let G be a connected Lie group, with Lie algebra . In 1977, Duflo constructed a homomorphism of -modules , which restricts to an algebra isomorphism on invariants. Kashiwara and Vergne (1978) proposed a conjecture on the Campbell-Hausdorff series, which (among other things) extends the Duflo theorem to germs of bi-invariant distributions on the Lie group G. The main results of the present paper are as follows. (1) Using a recent result of Torossian (2002), we establish the Kashiwara–Vergne conjecture for any Lie group G. (2) We give a reformulation of the Kashiwara–Vergne property in terms of Lie algebra cohomology. As a direct corollary, one obtains the algebra isomorphism , as well as a more general statement for distributions.     

Commutativity of the centralizer of a subalgebra in a universal enveloping algebra

Let G be a reductive algebraic group over an algebraically closed field of characteristic zero, and let \(\mathfrak{h}\) be an algebraic subalgebra of the tangent Lie algebra \(\mathfrak{g}\) of G. We find all subalgebras \(\mathfrak{h}\) that have no nontrivial characters and whose centralizers \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} \) and \(P(\mathfrak{g})^\mathfrak{h} \) in the universal enveloping algebra \(\mathfrak{U}(\mathfrak{g})\) and in the associated graded algebra \(P(\mathfrak{g})\), respectively, are commutative. For all these subalgebras, we prove that \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} = \mathfrak{U}(\mathfrak{h})^\mathfrak{h} \otimes \mathfrak{U}(\mathfrak{g})^\mathfrak{g} \) and \(P(\mathfrak{g})^\mathfrak{h} = P(\mathfrak{h})^\mathfrak{h} \otimes P(\mathfrak{g})^\mathfrak{g} \). Furthermore, we obtain a criterion for the commutativity of \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} \) in terms of representation theory.     

Arens-Michael enveloping algebras and analytic smash products

Let be a finite-dimensional complex Lie algebra, and let be its universal enveloping algebra. We prove that if , the Arens-Michael envelope of is stably flat over (i.e., if the canonical homomorphism is a localization in the sense of Taylor (1972), then is solvable. To this end, given a cocommutative Hopf algebra and an -module algebra , we explicitly describe the Arens-Michael envelope of the smash product as an ``analytic smash product' of their completions w.r.t. certain families of seminorms.

     

Invariant distributions supported on the nilpotent cone of a semisimple Lie algebra

Let be a semisimple complex Lie algebra with adjoint group and be the algebra of differential operators with polynomial coefficients on . If is a real form of , we give the decomposition of the semisimple -module of invariant distributions on supported on the nilpotent cone.

     

Extensions of Hopf Algebras and Lie Bialgebras

Let , be finite-dimensional Lie algebras over a field of characteristic zero. Regard and , the dual Lie coalgebra of , as Lie bialgebras with zero cobracket and zero bracket, respectively. Suppose that a matched pair of Lie bialgebras is given, which has structure maps . Then it induces a matched pair of Hopf algebras, where is the universal envelope of and is the Hopf dual of . We show that the group of cleft Hopf algebra extensions associated with is naturally isomorphic to the group of Lie bialgebra extensions associated with . An exact sequence involving either of these groups is obtained, which is a variation of the exact sequence due to G.I. Kac. If , there follows a bijection between the set of all cleft Hopf algebra extensions of by and the set of all Lie bialgebra extensions of by .

     

Cohomology and Support Varieties for Restricted Lie Superalgebras

Let $(\mathfrak{g}, [p]) $ be a restricted Lie superalgebra over an algebraically closed field k of characteristic p?>?2. Let $\mathfrak{u}(\mathfrak{g})$ denote the restricted enveloping algebra of $\mathfrak{g}$ . In this paper we prove that the cohomology ring $\operatorname{H}^\bullet(\mathfrak{u}(\mathfrak{g}), k)$ is finitely generated. This allows one to define support varieties for finite dimensional $\mathfrak{u}(\mathfrak{g})$ -supermodules. We also show that support varieties for finite dimensional $\mathfrak{u}(\mathfrak{g})$ - supermodules satisfy the desirable properties of a support variety theory.