Classification of Simple Cuspidal Modules over a Lattice Lie Algebra of Witt type

极小特征特征sheaves在分层李代数上的研究

Cuspidal character sheaves on graded Lie algebras

本文通过邻近循环构造和傅里叶-萨托变换，完成了文金类型I分层经典李代数极小特征特征sheaves的分类。

Cuspidal modules for the derivation Lie algebra over a rational quantum torus

Let $\mathbb C_Q$ denote a rational quantum torus with $d$ variables, and $\mathcal Z$ be the centre of $\mathbb C_Q$. In this paper we give a explicit description of the structure of the cuspidal modules for the derivation Lie algebra $\mathcal D$ over $\mathbb C_Q$, with an extra associative $\mathcal Z$-action.

Cuspidal representations of sl(n+1)

In this paper we study the subcategory of cuspidal modules of the category of weight modules over the Lie algebra sl(n+1). Our main result is a complete classification and explicit description of the indecomposable cuspidal modules.

秩≥1的超共形代数上的尖点模

Cuspidal modules over Superconformal algebras of rank \geq 1

本文对秩≥1的超共形代数及其中心扩张上的所有尖点模进行了显式分类，发现除K(4)的一个特定中心扩张外，尖点模的中心荷平凡，并影响了K(3)、CK(6)和K^(2)(4)的表示理论。

Blocks of the category of cuspidal $\mathfrak{sp}_{2n}$-modules

We show that every block of the category of cuspidal generalized weight modules with finite dimensional generalized weight spaces over the Lie algebra sp(2n)(C) is equivalent to the category of finite dimensional C[[t_1,t_2,...,t_n]]-modules.

关于由cuspidal李代数表示诱导的一些非主次局部解析表示

On some non-principal locally analytic representations induced by cuspidal Lie algebra representations

本文研究了由cuspidal李代数表示诱导的非主次局部解析表示的构造与性质，证明了其ind-admissible性及超cuspidal性，并在特定情况下证明了拓扑不可约性。

Finite weight modules over twisted affine Lie superalgebras

This work provides the first step toward the classification of irreducible finite weight modules over twisted affine Lie superalgebras. We study all such modules whether the canonical central element acts as a nonzero multiple of the identity map or not. Moreover, we reduce the classification of some subclasses of irreducible finite weight modules to the classification of cuspidal modules of finite dimensional cuspidal superalgebras which is known by a work of Dimitrov, Mathieu and Penkov.

映射与仿射李超代数的 Harish-Chandra 模

Harish-Chandra modules for map and affine Lie superalgebras

本文分类了基本经典映射超代数上具有有限权重的单模，证明其由尖点有界模抛物诱导且同构于赋值模，并将结果推广到仿射 Kac-Moody 李超代数，构造零级有界单模并给出分类猜想。

Quasi-integrable modules, a class of non-highest weight modules over twisted affine Lie superalgebras

In this paper, we characterize quasi-integrable modules, of nonzero level, over twisted affine Lie superalgebras. We show that quasi-integrable modules are not necessarily highest weight modules. We prove that each quasi-integrable module is parabolically induced from a cuspidal module, over a finite dimensional Lie superalgebra having a Cartan subalgebra whose corresponding root system just contain real roots; in particular, the classification of quasi-integrable modules is reduced to the known classification of cuspidal modules over such Lie superalgebras.

On the restricted Lie algebra structure for the Witt Lie algebra in finite characteristic

We show that the p-operator in the Witt algebra (the restricted Lie algebra of derivations of the quotient of the polynomial algebra over a field of characteristic p by the ideal generated by the p-th power of the indeterminant) is given by multiplication by a scalar.

On the Schur indices of cuspidal unipotent characters

In previous work of Gow, Ohmori, Lusztig and the author, the Schur indices of all unipotent characters of finite groups of Lie type have been explicitly determined except for six cases in groups of type $F_4$, $E_7$ and $E_8$. In this paper, we show that the Schur indices of all cuspidal unipotent characters for type $F_4$ and $E_8$ are~1, assuming that the group is defined over a field of ``good'' characteristic. This settles four out of the six open cases. For type $E_7$, we show that the Schur indices are at most~2.

Cuspidal characters and automorphisms

We investigate the action of outer automorphisms of finite groups of Lie type on their irreducible characters. We obtain a definite result for cuspidal characters. As an application we verify the inductive McKay condition for some further infinite families of simple groups at certain primes.

Simple Lie algebras which generalize Witt algebras

We introduce a new class of simple Lie algebras $W(n,m)$ that generalize the Witt algebra by using "exponential" functions, and also a subalgebra $W^*(n,m)$ thereof; and we show each derivation of $W^*(1,0)$ can be written as a sum of an inner derivation and a scalar derivation. The Lie algebra $W(n,m)$ is $Z$-graded and is infinite growth.

Cuspidal $\mathfrak{sl}_n$-modules and deformations of certain Brauer tree algebras

We show that the algebras describing blocks of the category of cuspidal weight (respectively generalized weight) $\mathfrak{sl}_n$-modules are one-parameter (respectively multi-parameter) deformations of certain Brauer tree algebras. We explicitly determine these deformations both graded and ungraded. The algebras we deform also appear as special centralizer subalgebras of Temperley-Lieb algebras or as generalized Khovanov algebras. They show up in the context of highest weight representations of the Virasoro algebra, in the context of rational representations of the general linear group and S…

On structure of graded restricted simple Lie algebras of Cartan type as modules over the Witt algebra

Any graded restricted simple Lie algebra of Cartan type contains a subalgebra isomorphic to the Witt algebra over a field of prime characteristic. As some analogue of study on branching rules for restricted non-classical Lie algebras, it is shown that each graded restricted simple Lie algebra of Cartan type can be decomposed into a direct sum of restricted baby Verma modules and simple modules as an adjoint module over the Witt algebra. In particular, the composition factors are precisely determined.

多项式代数上的仿射Nappi-Witten李代数的表示

Representations of affine Nappi-Witten Lie algebras over polynomial algebras

本文分类了Nappi-Witten李代数H4及其仿射代数上的秩一Cartan自由模，给出了不可约充要条件，并应用于仿射-Virasoro Nappi-Witten李代数。

Representations of spin quiver Hecke algebras for orthosymplectic Lie superalgebras

We construct all the irreducible representations of spin quiver Hecke algebras for orthosymplectic Lie superalgebras $osp(1|2n),$ and show that their highest weights are given by the dominant words. We use the dominant Lyndon words to construct the cuspidal modules and show that the irreducible representations are the simple heads of standard representations constructed by induction from the cuspidal modules.

有限李型群的幂单尖顶表示的Tate上同调

Tate cohomology of unipotent cuspidal representations of finite groups of Lie type

本文针对有限李型群的幂单尖顶表示，利用广义Gelfand-Graev表示与Harish-Chandra诱导构造的投射生成元的重数1性质，证明了特定Tate上同调群的不可约性与消失性。

Whittaker modules for the twisted affine Nappi-Witten Lie algebra $\widehat{H}_{4}[τ]$

The Whittaker module $M_ψ$ and its quotient Whittaker module $L_{ψ, ξ}$ for the twisted affine Nappi-Witten Lie algebra $\widehat{H}_{4}[τ]$ are studied. For nonsingular type, it is proved that if $ξ\neq 0$, then $L_{ψ,ξ}$ is irreducible and any irreducible Whittaker $\widehat{H}_{4}[τ]$-module of type $ψ$ with ${\bf k}$ acting as a non-zero scalar $ξ$ is isomorphic to $L_{ψ,ξ}$. Furthermore, for $ξ=0$, all Whittaker vectors of $L_{ψ, 0}$ are completely determined. For singular type, the Whittaker vectors of $L_{ψ, ξ}$ with $ξ\neq 0$ are fully characterized.

有限截断当前李代数上的简单模

Simple modules over truncated current Lie algebras of the Witt algebra

本文研究了有限截断当前李代数W_ℓ上p-特征为χ且高度不超过1的简单模，并给出其分类；进一步探讨了p-特征高度大于1的简单模家族。

Weight modules of D(2,1,a)

A weight module of a basic Lie superalgebra is called finite if all of its weight spaces are finite dimensional, and it is called bounded if there is a uniform bound on the dimension of a weight space. The minimum bound is called the degree of the module. For the basic Lie superalgebra D(2,1,a), we prove that every simple weight module is bounded and has degree less than or equal to 8. This bound is attained by a cuspidal module if and only if it is "typical". Atypical cuspidal modules have degree less than or equal to 6 and greater than or equal to 2.

Integrable modules for affine Lie superalgebras

Irreducible nonzero level modules with finite-dimensional weight spaces are studied for non-twisted affine Lie superalgebras. A complete classification is obtained for superalgebras A(m,n)^ and C(n)^. In other cases the classification problem is reduced to the classification of cuspidal modules over finite-dimensional cuspidal Lie superalgebras described by Dimitrov, Mathieu and Penkov. It is also shown that any irreducible weakly integrable (in the sense of Kac and Wakimoto) module is highest weight.

拟泊松模与Harish-Chandra AD-模

Quasi-Poisson Modules and Harish-Chandra AD-Modules

本文引入Lie-Rinehart对上的拟泊松模概念，证明特定Lie-Rinehart对上的单尖拟泊松模与单尖Harish-Chandra A𝔣-模一一对应，并分类了这些拟泊松模，表明每个这样的模是通过指定作用由可容许的𝔤𝔩(m+1,n)-模张量得到的。

Restricted Cohomology of Modular Witt Algebras

We classify one-dimensional restricted central extensions of the modular Witt Lie algebra in characteristic $p>3$.

映射全环面李代数上不可约Harish-Chandra模的分类

Classification of irreducible Harish-Chandra modules over map full toroidal Lie algebras

本文分类了映射全环面李代数上的不可约Harish-Chandra模，证明其必为尖点模或最高权模，且均为单点赋值模。

On Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras

We give a uniform proof of a significant part of the Cachazo-Douglas-Seiberg-Witten conjecture by using topology and geometry of infinite Grassmannians.

A Poincaré-Birkhoff-Witt Theorem for profinite pronilpotent Lie algebras

We prove a version of the Poincaré-Birkhoff-Witt Theorem for profinite pronilpotent Lie algebras in which their symmetric and universal enveloping algebras are replaced with appropriate formal analogues and discuss some immediate corollaries of this result.

Some representations of nongraded Lie algebras of generalized Witt type

In a paper by Xu, some simple Lie algebras of generalized Cartan type were constructed, using the mixtures of grading operators and down-grading operators. Among them, are the simple Lie algebras of generalized Witt type, which are in general nongraded and have no torus. In this paper, some representations of these simple Lie algebras of generalized Witt type are presented.

辛李代数的表示

Representations of Hamiltonian Lie algebras

本文研究了辛李代数到哈密顿李代数的表示映射，证明该映射在有限例外情况下保持不可约性，并展示了哈密顿李代数的cuspidal不可约模的权重多重性与辛李代数模的维数关系。

q-Witt Algebras, q-Virasoro algebra, q-Lie Algebras, q-Holomorph Structure and Representations

For q generic or a primitive l-th root of unity, q-Witt algebras are described by means of q-divided power algebras. The structure of the universal q-central extension of the q-Witt algebra, the q-Virasoro algebra, is also determined. q-Lie algebras are investigated and the q-PBW theorem for the universal enveloping algebras of q-Lie algebras is proved. A realization of a class of representations of the q-Witt algebras is given. Based on it, the q-holomorph structure for the q-Witt algebras is constructed, which interprets the realization in the context of representation theory.

Witt 李代数的多项式表示

Polynomial representations of the Witt Lie algebra

本文研究 Witt 李代数的多项式表示，证明有限生成表示是诺特的且具有有理 Hilbert 级数，并建立其与有限集合范畴及自同态幺半群表示的等价性，作为 Operad 化 Schur–Weyl 对偶的特例。

Infinite Dimensional 3-Lie Algebras and Their Connections to Harish-Chandra Modules

In this paper we construct two kinds of infinite-dimensional 3-Lie algebras from a given commutative associative algebra, and show that they are all canonical Nambu 3-Lie algebras. We relate their inner derivation algebras to Witt algebras, and then study the regular representations of these 3-Lie algebras and the natural representations of the inner derivation algebras. In particular, for the second kind of 3-Lie algebras, we find that their regular representations are Harish-Chandra modules, and the inner derivation algebras give rise to intermediate series modules of the Witt algebras and c…

Cohomology of Hom-Lie superalgebras and q-deformed Witt superalgebra

The purpose of this paper is to define the representation and the cohomology of Hom-Lie superalgebras. Moreover we study Central extensions and provide as application the computations of the derivations and second cohomology group of q-deformed Witt superalgebra.

On the Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras, II

Recently, motivated by supersymmetric gauge theory, Cachazo, Douglas, Seiberg, and Witten proposed a conjecture about finite dimensional simple Lie algebras, and checked it in the classical cases. Later V. Kac and the author proposed a uniform approach to this conjecture, based on the theory of abelian ideals in the Borel subalgebra; this allowed them to check the conjecture for type G_2. In this note we further develop this approach, and propose three natural conjectures which imply the three parts of the CDSW conjecture. In a sense, these conjectures explain why the CDSW conjecture should be…

Study of parity sheaves arising from graded Lie algebra

Let $G$ be a complex, connected, reductive, algebraic group, and $χ:\mathbb{C}^\times \to G$ be a fixed cocharacter that defines a grading on $\mathfrak{g}$, the Lie algebra of $G$. Let $G_0$ be the centralizer of $χ(\mathbb{C}^\times)$. In this paper, we study $G_0$-equivariant parity sheaves on $\mathfrak{g}_n$, under some assumptions on the field $\Bbbk$ and the group $G$. The assumption on $G$ holds for $GL_n$ and for any $G$, it recovers results of Lusztig in characteristic $0$. The main result is that every parity sheaf occurs as a direct summand of the parabolic induction of some cuspid…

分次李代数的不变系统与特征层

Invariant systems and character sheaves for graded Lie algebras

该研究通过引入微分方程不变系统构造分次李代数上的特征层，并与既有结果共同给出Vinberg I型经典情形的全部尖点特征层。

关于阶化李代数本原对的注记

A Note on primitive pairs for graded Lie algebras

本研究构建正特征域上Z-阶化李代数的本原对理论，证明不可分解宇称层可表为Levi子群本原数据诱导复形的直和项，揭示阶化李代数几何与表示论的深刻相容性，为正特征下阶化广义Springer对应奠定基础。

Quantizations of generalized-Witt algebra and of Jacobson-Witt algebra in the modular case

We quantize the generalized-Witt algebra in characteristic 0 with its Lie bialgebra structures discovered by Song-Su (\cite{GY}). Via a modulo p reduction and a modulo "p-restrictedness" reduction process, we get 2^n{-}1 families of truncated p-polynomial noncocommutative deformations of the restricted universal enveloping algebra of the Jacobson-Witt algebra \mathbf{W}(n;\underline{1}) (for the Cartan type simple modular restricted Lie algebra of W type). They are new families of noncommutative and noncocommutative Hopf algebras of dimension p^{1+np^n} in characteristic p. Our results general…

Poincare-Birkhoff-Witt expansions of the canonical elliptic differential form

We study the canonical U(\n)-valued elliptic differential form, whose projections to different Kac-Moody algebras are key ingredients of the hypergeometric integral solutions of elliptic KZ differential equations and Bethe ansatz constructions. We explicitly determine the coefficients of the projections in the simple Lie algebras A_r, B_r, C_r, D_r in a conveniently chosen Poincare-Birkhoff-Witt basis. As an application we give a new formula for eigenfunctions of Hamiltonians of the Calogero-Moser model.

Classification of simple strong Harish-Chandra $W(m,n)$-modules

We classify all simple strong Harish-Chandra modules for the Lie superalgebra $W(m,n)$. We show that every such module is either strongly cuspidal or a module of the highest weight type. We construct tensor modules for $W(m,n)$, which are parametrized by simple finite-dimensional $gl(m,n)$-modules and show that every simple strongly cuspidal $W(m,n)$-module is a quotient of a tensor module. Finally, we realize modules of the highest weight type as simple quotients of the generalized Verma modules induced from tensor modules for $W(m-1,n)$.

Witt与Virasoro代数上的分次反pre-Lie代数结构

Graded anti-pre-Lie algebraic structures on Witt and Virasoro algebras

本文通过分类Witt代数的不可分解权表示，给出了其上的分次反pre-Lie代数结构及同构分类，并证明了Virasoro代数上不存在满足自然条件的此类结构。

Non-weight representations of Cartan type S Lie algebras

For the two Cartan type S subalgebras of the Witt algebra $\W_n$, called Lie algebras of divergence-zero vector fields, we determine all module structures on the universal enveloping algebra of their Cartan subalgebra $\h_n$. We also give all submodules of these modules.

Singular Virasoro vectors and Lie algebra cohomology

We present an explicit formula for a new family of Virasoro singular vectors. As a corrolary we get formulas for differentials of Feigin-Fuchs-Rocha-Carridi-Wallach resolution of the the positive nilpotent part of Virasoro (or Witt) algebra $L_1$.

Filtering bases and cohomology of nilpotent subalgebras of Witt and $\widetilde{sl}_2$ Lie algebras

We study the cohomology with trivial coefficients of Lie algebras L_k of the polynomial vector fields on the line with zero $k$-jet, (k>=1), and the cohomology of the similar subalgebras {L}_k of the polynomial loops algebra $\widetilde{sl}_2$. In both cases we construct the special bases (filtering bases) in the external complexes of these algebras. A spectral sequence based on this construction allows to completely find the cohomology of L_k and {L}_k. We also apply the filtering bases to find the spectral resolution of the Laplace operators for algebras L_1 and L_0, and obtain explicit form…

分次李代数的特征层：稳定分次

Character Sheaves for Graded Lie Algebras: Stable Gradings

本研究针对稳定分次李代数，通过分析[GVX2]中邻近周期层的傅里叶变换，构造出推测为尖点特征层的满支撑特征层，涉及单位根处复反射群相关Hecke代数的不可约表示。

Simple weight modules with finite-dimensional weight spaces over Witt superalgebras

Let $A_{m,n}$ be the tensor product of the Laurient polynomial algebra in $m$ even variables and the exterior algebra in $n$ odd variables over the complex field $\bC$, and the Witt superalgebra $W_{m,n}$ be the Lie superalgebra of superderivations of $A_{m,n}$. In this paper, we classify the simple weight $W_{m,n}$ modules with finite-dimensional weight spaces with respect to the standard Cartan algebra of $W_{m,0}$. Every such module is either a simple quotient of a tensor module or a module of highest weight type.

Classification of $\mathfrak{sl}_3$ relations in the Witt group of nondegenerate braided fusion categories

The Witt group of nondegenerate braided fusion categories $\mathcal{W}$ contains a subgroup $\mathcal{W}_\text{un}$ consisting of Witt equivalence classes of pseudo-unitary nondegenerate braided fusion categories. For each finite-dimensional simple Lie algebra $\mathfrak{g}$ and positive integer $k$ there exists a pseudo-unitary category $\mathcal{C}(\mathfrak{g},k)$ consisting of highest weight integerable $\hat{g}$-modules of level $k$ where $\hat{\mathfrak{g}}$ is the corresponding affine Lie algebra. Relations between the classes $[\mathcal{C}(\mathfrak{sl}_2,k)]$, $k\geq1$ have been compl…

Weight modules for map (super)algebra related to the Virasoro algebra

We classify Jet modules for the Lie (super)algebras $\mathfrak{L}=W\ltimes(\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}])$, where $W$ is the Witt algebra and $\mathfrak{g}$ is a Lie superalgebra with an even diagonlizable derivation. Then we give a concept method to classify all simple cuspidal modules for $\mathfrak{L}$ and the map superalgebras, which are of the form $\mathfrak{L}\otimes R$, where $R$ is a Noetherian unital supercommutative associative superalgebra.

约化李代数上 D-模的广义 Springer 理论

Generalized Springer Theory for D-modules on a Reductive Lie Algebra

本文建立了约化李代数上等变 D-模范畴的广义 Springer 理论，将其正交分解为以尖点数据为指标的分块，并证明每个分块等价于中心上相对 Weyl 群等变 D-模范畴，核心方法是发展抛物诱导与限制函子及相应单子理论。

Imaginary Schur-Weyl duality

We study imaginary representations of the Khovanov-Lauda-Rouquier algebras of affine Lie type. Irreducible modules for such algebras arise as simple heads of standard modules. In order to define standard modules one needs to have a cuspidal system for a fixed convex preorder. A cuspidal system consists of irreducible cuspidal modules---one for each real positive root for the corresponding affine root system ${\tt X}_l^{(1)}$, as well as irreducible imaginary modules---one for each $l$-multipartition. We study imaginary modules by means of `imaginary Schur-Weyl duality'. We introduce an imagina…

Central extensions, derivations, and automorphisms of semi-direct sums of the Witt algebra with its intermediate series modules

Lie algebras formed via semi-direct sums of the Witt algebra $\text{Der}(\mathbb{C}[t,t^{-1}])$ and its modules have become increasingly prominent in both physics and mathematics in recent years. In this paper, we complete the study of (Leibniz) central extensions, derivations and automorphisms of the Lie algebras formed from the semi-direct sum of the Witt algebra and its indecomposable intermediate series modules (that is, graded modules with one-dimensional graded components). Our techniques exploit the internal grading of the Witt algebra, which can be applied to a wider class of graded Li…

Invariant Differential Operators for Non-Compact Lie Groups: Parabolic Subalgebras

In the present paper we start the systematic explicit construction of invariant differential operators by giving explicit description of one of the main ingredients - the cuspidal parabolic subalgebras. We explicate also the maximal parabolic subalgebras, since these are also important even when they are not cuspidal. Our approach is easily generalised to the supersymmetric and quantum group settings and is necessary for applications to string theory and integrable models.

Derived Poincaré-Birkhoff-Witt theorems (with an appendix by Vladimir Dotsenko)

We define derived Poincaré--Birkhoff--Witt maps of dg operads or derived PBW maps, for short, which extend the definition of PBW maps between operads of V.~Dotsenko and the second author in 1804.06485, with the purpose of studying the universal enveloping algebra of dg Lie algebras as a functor on the homotopy category. Our main result shows that the map from the homotopy Lie operad to the homotopy associative operad is derived PBW, which gives us an amenable description of the homology of the universal envelope of an $L_\infty$-algebra in the sense of Lada--Markl. We deduce from this several …

On Weakly Complete Universal Enveloping Algebras: A Poincare-Birkhoff-Witt Theorem

We study universal enveloping Hopf algebras of Lie algebras in the category of weakly complete vector spaces over the real and complex field.