On differentiability of vectors in Lie group representations
Ingrid Beltita, Daniel Beltita
本文讨论了局部凸空间上无限维李群表示中可微向量的线性问题，它类似于函数的方向导数的线性问题.
We address a linearity problem for differentiable vectors in representations of infinite-dimensional Lie groups on locally convex spaces, which is similar to the linearity problem for the directional derivatives of functions.
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On Differentiable Vectors for Representations of Infinite Dimensional Lie Groups
在本文中，我们开发了两种类型的工具来处理的连续表示$π\: G \to \GL(V)$的无限维李群$G$局部凸空间$V$上的向量的可微性。第一类结果涉及光滑向量空间$V^\infty$。如果$G$是一个Banach-李群，我们定义一个拓扑空间$V^\infty$上的光滑向量的$G$在这个空间上的作用是光滑的。如果$V$是Banach空间，则$V^\infty$是Fréchet空间。这特别适用于$C^*$ -动力系统$(\cA,G, α)$，其中$G$是一个Banach-李群。对于酉表示，我们证明了一个向量$v$是光滑的，如果相应的正定函数$\la π(g)v,v\ra$是光滑的。第二类结果涉及标准的$C^k$ -向量的运营商的派生代表的Banach-李群$G$作用于Banach空间$V$。特别是，我们为每个$k \in \N$提供连续酉表示的例子，其中$C^{k+1}$ -向量的空间是平凡的，并且$C^k$ -向量的空间是稠密的。
Measurable regularity properties of infinite-dimensional Lie groups
我们考虑李群G（可能是无限维的）上的形式为y '（t）=f（t，y（t））的微分方程，其中f：[0，1] x G -> TG是时间依赖的左不变向量场，它对t具有可测的（但不一定是连续的）依赖性.如果G的李代数g中对应于L^1-曲线c：[0，1] -> g的每个f都存在从G的中性元e开始的关于[0，1]的解Evol（c）：=y，并且Evol作为从L^1（[0，1]，g）到C（[0，1]，G）的映射是光滑的，则G称为L^1-正则的.我们证明了所有的Banach-李群都是L^1-正则的，以及有限维李群的所有直接极限和无限维李群的许多进一步的经典例子，如仿紧有限维光滑流形的自同态群。如果李群G是L^1-正则的，则Trotter乘积公式和换位子公式在强意义下在G中成立。如果G仅满足某些较弱的可测正则性（如L^p-正则性），则同样的结论成立，并讨论了这些性质。作为工具，我们研究了向量值Lebesgue空间和向量值绝对连续函数空间的某些非线性映射的可微性。
Invariant Hilbert spaces of distribution vectors in Lie group representations
本文证明了李群的酉不可约表示中，分布向量的唯一非零不变希尔伯特空间即为该表示的希尔伯特空间。
Strong factorization theorem for smooth vectors of exponential solvable Lie group representations
本文改进了Dixmier-Malliavin定理，为指数可解李群在弗雷歇空间上的光滑向量建立了新的强因子分解性质。
Strong factorization of ultradifferentiable vectors associated with compact Lie group representations
本文针对紧李群表示的超光滑向量，证明了强分解定理并解决了Gimperlein等人的猜想。
In this paper, we established the relationship among left-invariant flat connections on Lie groups, left-symmetric algebras, Frobenius Lie algebras and étale prehomogeneous vector spaces, gave a one-to-one correspondence between the left-symmetric Lie algebras with a right identity and the étale prehomogeneous vector spaces for a Lie group, and proved that, in essence, any left-symmetric structure on a reductive Lie algebra has a right identity, which implies that the classification of flat connections on a reductive Lie group $G$ amounts to that of étale prehomogeneous vector spaces for $G$. …
How to realize Lie algebras by vector fields
给出了一种将有限维李代数嵌入向量场的李代数（以及将李超代数嵌入向量场的李超代数）的算法，该算法适用于任何特征的基域.该算法通过再现Cartan对G（2）的李代数的解释作为保持某些不可积分布的李代数来说明。类似的算法和解释也适用于其他例外单李代数，以及所有非例外单李代数和许多非例外单李代数，以及许多李超代数。
Irreducible Modules for the Lie Algebra of Divergence Zero Vector Fields on a Torus
研究了环面上散度零向量场李代数的某些表示的不可约性。在“d维环面的同构的李代数的不可约表示”中，S. Eswara Rao构造了d维环面上多项式向量场的李代数的模，并确定了不可约的条件.本文考虑了这些模对散度零向量场的子代数的限制。本文证明了Rao的结果对于发散零向量场的李代数可转化为类似的不可约条件。
Realization by linear vector fields is constructed for any Lie algebra which admits a biorthogonal system and for its any suitable representation. The embedding into Lie algebras of linear vector fields is analogous to the classical Jordan-Schwinger map. A number of examples of such Lie algebras of linear vector fields is computed. In particular, we obtain examples of the twisted Heisenberg-Virasoro Lie algebra and the Schrödinger-Virasoro Lie algebras among others. More generally, we construct an embedding of an arbitrary locally convex topological algebra into the Cuntz algebra.
Differential equations in vertex algebras and simple modules for the Lie algebra of vector fields on a torus
本文研究了二维环面上向量场的李代数的不可约表示，这些不可约表示是利用广义韦尔马模构造的。我们证明，对于一定的参数选择，这些表示在限制于向量场李代数中的循环子代数时仍然不可约。我们证明了这一结果，通过研究顶点代数与李代数的向量场的环面和解决非交换微分方程，我们得出使用顶点代数技术。
Maximal Graded Solvable Subalgebras of Simple Finite-Dimensional Vectorial Lie Superalgebras
本文在复数域上，对标准分次下所有简单有限维向量李超代数的极大分次可解子代数进行了完整分类。
For a smooth irreducible affine algebraic variety we study a class of gauge modules admitting compatible actions of both the algebra $A$ of functions and the Lie algebra $\mathcal{V}$ of vector fields on the variety. We prove that a gauge module corresponding to a simple $\mathfrak{gl}_N$-module is irreducible as a module over the Lie algebra of vector fields unless it appears in the de Rham complex.
We reprove the results of Jordan [18] and Siebert [31] and show that the Lie algebra of polynomial vector fields on an irreducible affine variety X is simple if and only if X is a smooth variety. Given proof is self-contained and does not depend on papers mentioned above. Besides, the structure of the module of polynomial functions on an irreducible smooth affine variety over the Lie algebra of vector fields is studied. Examples of Lie algebras of polynomial vector fields on an N-dimensional sphere, non-singular hyperelliptic curves and linear algebraic groups are considered.
Graded Lie algebras and regular prehomogeneous vector spaces with one-dimensional scalar multiplication
研究了一维标量乘法正则约化PV与分次李代数结构之间的关系。我们将证明这种PV的正则性是由分次李代数的$\mathfrak{sl}_2$ -三元组描述的。
Defining relations for classical Lie algebras of polynomial vector fields
本文明确地描述了多项式系数向量场的单李代数及其无散度、Hamilton和切触向量场的子代数和Poisson代数（在多项式上实现）的定义关系.我们考虑这些李代数的生成元对应于与这些代数的标准分次相关联的单根系统。（这些系统的单根区别在意义上的Penkov和Serganova。
Singular Virasoro vectors and Lie algebra cohomology
我们提出了一个新的Virasoro奇异向量族的显式公式。作为推论，我们得到了Virasoro（或Witt）代数$L_1$的正幂零部分的Feigin-Fuchs-Rocha-Carridi-Wallach分解的微分公式.
The classification of simple complex Lie superalgebras of polynomial vector fields and their deformations
本文综述多项式增长的单无限维复$\u2124$-分次李（超）代数及其形变分类，修正已有结果，解决W-分次与W-滤过单向量李超代数的分类等问题，并提出相关猜想与方法应用展望。
The Lie algebra generated by $m\ $ $p$-dimensional Grassmannian Dirac operators and $m\ $ $p$-dimensional vector variables is identified as the orthogonal Lie algebra $\mathfrak{so}(2m+1)$. In this paper, we study the space $\mathcal{P}$ of polynomials in these vector variables, corresponding to an irreducible $\mathfrak{so}(2m+1)$ representation. In particular, a basis of $\mathcal{P}$ is constructed, using various Young tableaux techniques. Throughout the paper, we also indicate the relation to the theory of parafermions.
For a simple Lie superalgebra of type BDFG, we give explicit formulas for singular vectors in a Verma module of highest weight $λ- ρ$, which have weight $s_γλ- ρ$ for certain positive non-isotropic roots $γ.$ This implies the existence of a nonzero homomorphism between the corresponding Verma modules.
An approach to annihilators in the context of vector field Lie algebras
本文提出一种描述李代数模零化子的一般方法，并将其用于向量场李代数张量模，高效证明了 Vec(R) 有界不可约模零化子的已有结论。
Let $W_n^+$ be the Lie algebra of the Lie algebra of vector fields on $\C^n$. In this paper, we classify all simple bounded weight $W_n^+$ modules. Any such module is isomorphic to the simple quotient of a tensor module $F(P,M)=P\otimes M$ for a simple weight module $P$ over the Weyl algebra $K_n^+$ and a finite dimensional simple $\gl_n$ module $M$.
Finite-dimensional subalgebras of a Lie algebra of smooth vector fields on a circle, as well as piecewise-smooth global transformations of a circle on itself, are considered. A canonical forms of realizations of two- and three-dimensional noncommutative algebras are obtained. It is shown that all other realizations of smooth vector fields are reduced to this form using global transformations. Some combinatorial formulas for the number of inequivalent realizations of these algebras are obtained.
Let ${\mathcal W}_n$ be the Lie algebra of polynomial vector fields. We classify simple weight ${\mathcal W}_n$-modules $M$ with finite weight multiplicities. We prove that every such nontrivial module $M$ is either a tensor module or the unique simple submodule in a tensor module associated with the de Rham complex on $\mathbb C^n$.
A brief proof of Lie's classification of solvable algebras of vector fields on the plane is given. The proof uses basic representation theory and PDEs.
We use the language of Lie pseudoalgebras to gain information about the representation theory of the simple infinite-dimensional linearly compact Lie superalgebra of exceptional type $E(5,10)$. This technology allows us to prove that the degree of singular vectors in minimal Verma modules is $\leq 14$. A few technical adjustments allow us to refine the bound, proving that the degree must always be $\leq 12$ and it is actually, except for a finite number of cases, $\leq 10$.
In this article, we exploit the theory of graded module category with semi-infinite character developed by Soergel in \cite{Soe} to study representations of the infinite dimensional Lie algebras of vector fields $W(n), S(n)$ and $H(n)$ $(n\geq 2)$, and obtain the description of indecomposable tilting modules. The character formulas for those tilting modules are determined.
We establish a closed formula for a singular vector of weight $λ-β$ in the Verma module of highest weight $λ$ for Lie superalgebra $\mathfrak{gl}(m|n)$ when $λ$ is atypical with respect to an odd positive root $β$. It is further shown that this vector is unique up to a scalar multiple, and it descends to a singular vector, again unique up to a scalar multiple, in the corresponding Kac module when both $λ$ and $λ-β$ are dominant integral.
Entrelacement d'algèbres de Lie [Wreath products for Lie algebras]
给出了任意两个李代数的圈积的定义和构造方法，希望能由此引出一个合适的李群是两个给定李群的圈积的定义.在这一过程中，需要引入许多新的概念。例如：形式级数，变量在一个向量空间中，系数在其他向量空间中。一个形式级数相对于另一个形式级数的推导。向量空间的李代数。向量空间上李代数的形式作用。一个李代数在自身上的基本形式作用（作为李群子的无穷小运算律的解析形式）。更一般地，两个李代数的圈积被定义，相对于第二个到任意向量空间上的形式作用。主要特点是：关于积向量空间上的圈积的三角作用的描述，以及一个Kaloujnine-Krasner型定理：本质上，它说的是一个给定的李代数被另一个李代数的所有李扩张确实是它们的圈积的子代数。
Lie's classification of finite dimensional algebras of Vector Fields in C^N
本文概述了Lie关于两变量和三变量向量场有限维子代数的经典结果的简要证明，并给出了任意N维C^N中最大秩向量场代数的结果。
Smooth vectors and Weyl-Pedersen calculus for representations of nilpotent Lie groups
给出了幂零李群酉不可约表示的光滑向量的一些最新结果。讨论了符号在余伴轨道上的伪微分算子在Weyl-Pedersen演算中的应用。
VB-groupoids are vector bundles in the category of Lie groupoids. They encompass several classical objects, including Lie group representations and 2-vector spaces. Moreover, they provide geometric pictures for 2-term representations up to homotopy of Lie groupoids. We attach to every VB-groupoid a cochain complex controlling its deformations and discuss its fundamental features, such as Morita invariance and a van Est theorem. Several examples and applications are given.
Contact Lie algebras of vector fields on the plane
本文给出了一维子流形的第一喷流空间上切触向量场的实李代数的完全分类。这完成了索菲斯李的分类所有可能的李代数的接触对称性的常微分方程。作为主要工具，我们使用过滤和分次李代数的抽象理论。我们还描述了文中所发现的新李代数的所有微分和积分不变量，并讨论了无限维情形。
We investigate the second cohomology space of the Lie superalgebra $\mathcal{K}(2)$ with coefficients in the superspace of weighted densities on the (1,2)-dimentional real superspace. We explicitly give cocycles spanning this cohomology space.
Differentiable vectors and unitary representations of Frechet-Lie supergroups
对于具有Trotter性质的局部凸李群，证明了酉表示的k次可微向量空间等于李代数作用的k重积整环的交.这个结果也适用于可度量化局部凸空间上局部指数群的连续表示。作为应用，我们将李超群表示论中的一个关键稳定性定理扩展到Banach情形之外。
On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups
设$G$是一个1-连通的Banach-李群，或者更一般地，是一个BCH-李群.在复包络代数$U_\C(\g)$上，定义了解析泛函的概念，证明了正解析泛函$λ$是可积的，对于$G$的酉表示的解析向量$v$，它具有$λ(D) = \la \ddπ(D)v, v\ra$的形式.在得到这个结果的过程中，我们得到了无界算子的无限维李代数的 *-表示到酉群表示的可积性准则。对于解析Fréchet-李群$G$的酉表示中向量$v$的矩阵系数$π^{v,v}(g) = \la π(g)v,v\ra$，我们证明了$v$是解析向量当且仅当$π^{v,v}$在单位邻域中解析.结合正解析泛函的结果，我们得到了1-连通Fréchet-BCH- Lie群$G$上的每一个局部正定解析函数都扩张为整体解析函数.
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