Strongly typical representations of the basic classical Lie superalgebras

Minimal W-superalgebras and modular representations of basic Lie superalgebras

Let $\mathfrak{g}=\mathfrak{g}_{\bar 0}+\mathfrak{g}_{\bar 1}$ be a basic Lie superalgebra over $\mathbb{C}$, and $e$ a minimal nilpotent element in $\mathfrak{g}_{\bar 0}$. Set $W_χ'$ to be the refined $W$-superalgebra associated with the pair $(\mathfrak{g},e)$, which is called a minimal $W$-superalgebra. In this paper we present a set of explicit generators of minimal $W$-superalgebras and the commutators between them. In virtue of this, we show that over an algebraically closed field $\mathds{k}$ of characteristic $p\gg0$, the lower bounds of dimensions in the modular representations of ba…

Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras

In this paper, we use geometrical methods adapted from the Borel-Weil-Bott theory to compute the character of every finite dimensional simple module over a basic classical Lie superalgebra.

Centralizers of nilpotent elements in basic classical Lie superalgebras in good characteristic

Let \mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}} be a basic classical Lie superalgebra over an algebraically closed field \mathbb{K} whose characteristic p>0 is a good prime for \mathfrak{g}. Let G_{\bar{0}} be the reductive algebraic group over \mathbb{K} such that \mathrm{Lie}(G_{\bar{0}})=\mathfrak{g}_{\bar{0}}. Suppose e\in\mathfrak{g}_{\bar{0}} is nilpotent. Write \mathfrak{g}^{e} for the centralizer of e in \mathfrak{g} and \mathfrak{z}(\mathfrak{g}^{e}) for the centre of \mathfrak{g}^{e}. We calculate a basis for \mathfrak{g}^{e} and \mathfrak{z}(\mathfrak{g}^{e}) by …

Whittaker categories and strongly typical Whittaker modules for Lie superalgebras

Following analogous constructions for Lie algebras, we define Whittaker modules and Whittaker categories for finite-dimensional simple Lie superalgebras. Results include a decomposition of Whittaker categories for a Lie superalgebra according to the action of an appropriate sub-superalgebra; and, for basic classical Lie superalgebras of type I, a description of the strongly typical simple Whittaker modules.

Finite $W$-superalgebras and the dimensional lower bounds for the representations of basic Lie superalgebras

In this paper we formulate a conjecture about the minimal dimensional representations of the finite $W$-superalgebra $U(\mathfrak{g}_\bbc,e)$ over the field of complex numbers and demonstrate it with examples including all the cases of type $A$. Under the assumption of this conjecture, we show that the lower bounds of dimensions in the modular representations of basic Lie superalgebras are attainable. Such lower bounds, as a super-version of Kac-Weisfeiler conjecture, were formulated by Wang-Zhao in \cite{WZ} for the modular representations of a basic Lie superalgebra ${\ggg}_{\bbk}$ over an a…

Support varieties and representations of tame basic classical Lie superalgebras

Let k be an algebraically closed field of characteristic p> 0. For a basic classical Lie superalgebra, we show its restricted cohomology algebra is a finitely generated algebra. Thus the cohomological support theory can be established. As a consequence, we show that the restricted enveloping algebra of a basic classical Lie superalgebra g is always wild except g=sl(2) or g=osp(1|2) or g=C(2). Moreover, all finite dimensional indecomposable restricted representations of u(osp(1|2)), the restricted enveloping algebra of Lie superalgebra osp(1|2), are determined.

Grothendieck rings of basic classical Lie superalgebras

The Grothendieck rings of finite dimensional representations of the basic classical Lie superalgebras are explicitly described in terms of the corresponding generalised root systems. We show that they can be interpreted as the subrings in the weight group rings invariant under the action of certain groupoids called Weyl groupoids.

Finite W-superalgebras for basic Lie superalgebras

We consider the finite $W$-superalgebra $U(\mathfrak{g_\bbf},e)$ for a basic Lie superalgebra ${\ggg}_\bbf=(\ggg_\bbf)_\bz+(\ggg_\bbf)_\bo$ associated with a nilpotent element $e\in (\ggg_\bbf)_{\bar0}$ both over the field of complex numbers $\bbf=\mathbb{C}$ and over $\bbf={\bbk}$ an algebraically closed field of positive characteristic. In this paper, we mainly present the PBW theorem for $U({\ggg}_\bbf,e)$. Then the construction of $U({\ggg}_\bbf,e)$ can be understood well, which in contrast with finite $W$-algebras, is divided into two cases in virtue of the parity of $\text{dim}\,\mathfra…

On finite dimensional representations of finite W-superalgebras

Let $\mathfrak{g}=\mathfrak{g}_{\bar{0}}+\mathfrak{g}_{\bar{1}}$ be a basic Lie superalgebra, $\mathcal{W}_0$ (resp.$\mathcal{W}$) be the finite W-(resp.super-) algebras constructed from a fixed nilpotent element in $\mathfrak{g}_{\bar{0}}$. Based on a relation between finite W-algebra $\mathcal{W}_0$ and W-superalgebra $\mathcal{W}$ found recently by the author and Shu, we study the finite dimensional representations of finite W-superalgebras in this paper. We first formulate and prove a version of Premet's conjecture for the finite W-superalgebras from basic simple Lie superalgebras. As in t…

Inverses of Cartan Matrices of Lie Algebras and Lie Superalgebra

We derive explicit formulas for the inverses of the Cartan matrices of the simple Lie algebras and the basic classical Lie superalgebras, as well as for their infinite generalizations.

Denominator identities for finite-dimensional Lie superalgebras and Howe duality for compact dual pairs

We provide formulas for the denominator and superdenominator of a basic classical type Lie superalgebra for any set of positive roots. We establish a connection between certain sets of positive roots and the theory of reductive dual pairs of real Lie groups. As an application of our formulas, we recover the Theta correspondence for compact dual pairs. Along the way we give an explicit description of the real forms of basic classical type Lie superalgebras.

Bounded weight modules for basic classical Lie superalgebras at infinity

We classify simple bounded weight modules over the complex simple Lie superalgebras $\mathfrak{sl}(\infty |\infty)$ and $\mathfrak{osp} (m | 2n)$, when at least one of $m$ and $n$ equals $\infty$. For $\mathfrak{osp} (m | 2n)$ such modules are of spinor-oscillator type, i.e., they combine into one the known classes of spinor $\mathfrak{o} (m)$-modules and oscillator-type $\mathfrak{sp} (2n)$-modules. In addition, we characterize the category of bounded weight modules over $\mathfrak{osp} (m | 2n)$ (under the assumption $\dim \, \mathfrak{osp} (m | 2n) = \infty$) by reducing its study to alread…

A new class of modules for Toroidal Lie Superalgebras

In this paper we construct a large class of modules for toroidal Lie superalgebras. Toroidal Lie superalgebras are universal central extension of G tensor A where G is a basic classical Lie superalgebra and A is a Laurent polynomial ring in sevaral variables. The case where G is simple finite dimensional Lie algebra is included.

超对称空间上的 Ghost 分布 II：基本经典李超代数

Ghost distributions on supersymmetric spaces II: basic classical superalgebras

本文针对基本经典李超代数研究超对称空间上的 ghost 分布，通过交错对构造并实现 ghost 代数，同时分析 Harish-Chandra 同态的单射性及其像。

The primitive spectrum of basic classical Lie superalgebras

We prove Conjecture 5.7 in [arXiv:1409.2532], describing all inclusions between primitive ideals for the general linear superalgebra in terms of the Ext1-quiver of simple highest weight modules. For arbitrary basic classical Lie superalgebras, we formulate two types of Kazhdan-Lusztig quasi-orders on the dual of the Cartan subalgebra, where one corresponds to the above conjecture. Both orders can be seen as generalisations of the left Kazhdan-Lusztig order on Hecke algebras and are related to categorical braid group actions. We prove that the primitive spectrum is always described by one of th…

Leibniz superalgebras and central extensions

Dialgebras are generalizations of associative algebras which give rise to Leibniz algebras instead of Lie algebras. In this paper we study super dialgebras and Leibniz superalgebras, which are $\z_2$-graded dialgebras and Leibniz algebras. We also study universal central extesions of Leibniz superalgebras and obtain some results as in the case of Lie superalgebras. We determine universal central extensions of basic classical Lie superalgebras in the category of Leibniz superalgebras. They play a key role in studying Leibniz superalgebras graded by finite root systems.

The generalized Kac-Wakimoto conjecture and support varieties for the Lie superalgebra osp(m|2n)

Atypicality is a fundamental combinatorial invariant for simple supermodules of a basic Lie superalgebra. Boe, Nakano, and the author gave a conjectural geometric interpretation of atypicality via support varieties. Inspired by low dimensional topology, Geer, Patureau-Mirand, and the author gave a generalization of the Kac-Wakimoto atypicality conjecture. We prove both of these conjectures for the Lie superalgebra osp(m|2n).

基本经典李超代数的最高权模的Gelfand--Kirillov维数

Gelfand--Kirillov dimensions of highest weight modules for basic classical Lie superalgebras

本文开发了一种组合算法，用于计算基本经典李超代数的单最高权模的GK维数，并给出了类型sl(m|n)和osp(2|2n)的显式公式，证明GK维数完全由李超代数的偶部决定。

奇特征下基本经典李超代数的Zassenhaus簇与其纯偶约化李子代数的双有理等价性

Birational equivalence of the Zassenhaus varieties for basic classical Lie superalgebras and their purely-even reductive Lie subalgebras in odd characteristic

本文研究奇特征代数闭域上基本经典李超代数的泛包络代数中心结构，证明其分式域与纯偶约化李子代数泛包络代数中心的分式域同构，进而得到两类Zassenhaus簇的双有理等价性及中心谱的有理性。

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。

超球面等变切片与基本经典李超代数

Hyperspherical equivariant slices and basic classical Lie superalgebras

本文完成了约化群超球面等变切片的分类，并证明该分类与基本经典李超代数的分类之间存在S-对偶关系。

Takiff 超代数的典型表示

Typical representations of Takiff superalgebras

本文研究 Takiff 超代数的典型表示，引入奇反射并给出典型表示的一般概念，完整描述 I 型有限维模特征标，证明 Kac 归纳函子的范畴等价，并分类非奇异 Whittaker 模。

Representations of Lie superalgebras in prime characteristic III

For a restricted Lie superalgebra g over an algebraically closed field of characteristic p > 2, we generalize the deformation method of Premet and Skryabin to obtain results on the p-power and 2-power divisibility of dimensions of g-modules. In particular, we give a new proof of the Super Kac-Weisfeiler conjecture for basic classical Lie superalgebras. The new proof allows us to improve optimally the assumption on p. We also establish a semisimplicity criterion for the reduced enveloping superalgebras associated with semisimple p-characters for all basic classical Lie superalgebras using the t…

The primitive spectrum and category O for the periplectic Lie superalgebra

We solve two problems in representation theory for the periplectic Lie superalgebra pe(n), namely the description of the primitive spectrum in terms of functorial realisations of the braid group and the decomposition of category O into indecomposable blocks. To solve the first problem we establish a new type of equivalence between category O for all (not just simple or basic) classical Lie superalgebras and a category of Harish-Chandra bimodules. The latter bimodules have a left action of the Lie superalgebra but a right action of the underlying Lie algebra. To solve the second problem we esta…

On Kostant's theorem for the Lie superalgebra Q(n)

In this paper we study finite W-algebras for basic classical superalgebras and Q(n) associated to the regular even nilpotent coadjoint orbits. We prove that this algebra satisfies the Amitsur-Levitzki identity and therefore all its irreducible representations are finite-dimensional. In the case of Q(n) we give an explicit description of the W-algebra in terms of generators and relation and realize it as a quotient of the super-Yangian of Q(1).

Šapovalov elements for simple Lie algebras and basic classical simple Lie superalgebras

Let $M(\gl)$ be a Verma module for a basic classical simple Lie superalgebra $\fg \neq G(3)$ defined using the distinguished Borel subalgebra, and let $\gc$ be an isotropic positive root of $\fg.$ As a special case of our first main result we show that if $μ, \gl \in \fh^*$ with $\gl-μ= \gc$ we have $$\dim \Hom_{\sfg}(M(μ),M(\gl))\le 1.$$ This result applies to the construction of Šapovalov elements for isotropic roots. The proof rests on a comparison with the corresponding result for a certain simple Lie algebra $G$.

On Kostant Root Systems for Lie Superalgebras

We study the eigenspace decomposition of a basic classical Lie superalgebra under the adjoint action of a toral subalgebra, thus extending results of Kostant. In recognition of Kostant's contribution we refer to the eigenspaces appearing in the decomposition as Kostant roots. We then prove that Kostant root systems inherit the main properties of classical root systems. Our approach is combinatorial in nature and utilizes certain graphs naturally associated with Kostant root systems. In particular, we reprove Kostant's results without making use of the Killing form.

Yangian of the general linear Lie superalgebra

We prove several basic properties of the Yangian of the general linear Lie superalgebra.

基本李超代数的旗超流形的平坦退化

Flat degenerations of flag supermanifolds for basic Lie superalgebras

本文将经典旗簇退化方法推广至超设定，通过调整表示基与滤波技术，构造了基本李超代数旗超流形的平坦退化，并探讨了退化至环面超簇的条件。

例外经典李超代数的超韦尔群的Coxeter图

Coxeter Graphs for Super Weyl Groups of Exceptional Classical Lie Superalgebras

本文研究例外经典李超代数的基本系统，确定对应超韦尔群的Coxeter图，验证了CLS中关于D(2,1,α)、F(4)和G(3)超韦尔群Coxeter图的猜想。

三次Dirac算子与基本经典李超代数的Dirac上同调

Cubic Dirac Operators and Dirac Cohomology for Basic Classical Lie Superalgebras

本文研究了基本经典李超代数上超模的Dirac上同调，通过三次Dirac算子建立了超模的Casselman-Osborne定理超代数版本，并证明了最高权超模的Dirac上同调非平凡，同时探讨了其与Kostant上同调的关系。

Typical blocks of Lie superalgebras in prime characteristic

For a type I basic classical Lie superalgebra $\mathfrak{g}=\mathfrak{g}_{\bar{0}} \oplus \mathfrak{g}_{\bar{1}}$, we establish an equivalence between typical blocks of categories of $U_χ(\mathfrak{g})$-modules and $U_χ(mathfrak{g}_{\bar{0})$-modules. We then deduce various consequences from the equivalence.

Representations of affine superalgebras and mock theta functions III

We study modular invariance of normalized supercharacters of tame integrable modules over an affine Lie superalgebra, associated to an arbitrary basic Lie superalgebra $ \mathfrak{g}. $ For this we develop a several step modification process of multivariable mock theta functions, where at each step a Zwegers' type "modifier" is used. We show that the span of the resulting modified normalized supercharacters is $ SL_2(\mathbb{Z}) $-invariant, with the transformation matrix equal, in the case the Killing form on $\mathfrak{g}$ is non-degenerate, to that for the subalgebra $ \mathfrak{g}^! $ of $…

PBW degenerations of Lie superalgebras and their typical representations

We introduce the PBW degeneration for basic classical Lie superalgebras and construct for all type I, $\mathfrak{osp}(1,2n)$ and exceptional Lie superalgebras new monomial bases. These bases are parametrized by lattice points in convex lattice polytopes, sharing useful properties such as the integer decomposition property. This paper is the first step towards extending the framework of PBW degenerations to the Lie superalgebra setting.

Hilbert series of typical representations for Lie superalgebras

Let g be a basic classical Lie superalgebra over C. In the case of a typical weight whose every nonnegative integer multiple is also typical, we compute a closed form for the Hilbert series whose coefficients encode the dimensions of finite-dimensional irreducible typical g-representations. We give a formula for this Hilbert series in terms of elementary symmetric polynomials and Eulerian polynomials. Additionally, we show a simple closed form in terms of differential operators.

Representations of affine Lie superalgebras and their quantization in type A

We construct a new family of irreducible modules over any basic classical affine Kac-Moody Lie superalgebra which are induced from modules over the Heisenberg subalgebra. We also obtain irreducible deformations of these modules for the quantum affine superalgebra of type A.

奇异经典李超代数的模表示与第一个超Kac-Weisfeiler猜想

Modular representations of strange classical Lie superalgebras and the first super Kac-Weisfeiler conjecture

本文初步研究了periplectic李超代数的模表示，并验证了关于其不可约模最大维数的第一个超Kac-Weisfeiler猜想。

Restricted Kac modules of Hamiltonian Lie superalgebras of odd type

This paper aims to describe the restricted Kac modules of restricted Hamiltonian Lie superalgebras of odd type over an algebraically closed field of characteristic $p>3$. In particular, a sufficient and necessary condition for the restricted Kac modules to be irreducible is given in terms of typical weights.

Serre functors for Lie algebras and superalgebras

We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category $\mathcal{O}$ associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to classical Lie superalgebras in many cases. Along the way we prove that category $\mathcal{O}$ and its parabolic generalizations for classical Lie superalgebras are categories with full projective functors. As an application we prove that in many cases the endomorphism algebra of the basic projective-injective module in (parabolic) category $\mathcal{O}$ for c…

On the Ghost Centre of Lie Superalgebras

We define a notion of ghost centre of a Lie superalgebra g=g_0+g_1 which is a sum of invariants with respect to the usual adjoint action (centre) and invariants with respect to a twisted adjoint action (``anticentre''). We calculate the anticentre in the case when the top external degree of g_1 is a trivial g_0-module. We describe the Harish-Chandra projection of the ghost centre for basic classical Lie superalgebras and show that for these cases the ghost centre coincides with the centralizer of the even part of the enveloping algebra. The ghost centre of a Lie superalgebra plays a role of th…

模李超代数的不可约模与第一Kac-Weisfeiler猜想的超版本

Irreducible modules of modular Lie superalgebras and super version of the first Kac-Weisfeiler conjecture

本文针对特征p>2的代数闭域上有限维限制李超代数，提出第一Kac-Weisfeiler猜想的超推广猜想，证明其对基本典型李超代数和完全可解限制李超代数成立，并研究了可解李超代数的不可约表示。

李超代数的Iwasawa分解

Iwasawa Decomposition for Lie Superalgebras

本文证明基本单李超代数上的任一保型对合或其与典范分次自同构的复合允许Iwasawa分解，利用广义根系及半单自同构环面的中心化子结果。

关于李超代数表示张量积的研究

On tensor products of representations of Lie superalgebras

本文研究基本经典单李超代数典型表示张量积的唯一因子分解条件，并在额外假设下证明若干系列单非典型不可约模的张量积唯一因子分解，从而给出 Rajan 结果的李超代数类比。

Bernstein-Gelfand-Gelfand resolutions for basic classical Lie superalgebras

We study Kostant cohomology and Bernstein-Gelfand-Gelfand resolutions for finite dimensional representations of basic classical Lie superalgebras and reductive Lie superalgebras based on them. For each choice of parabolic subalgebra and irreducible representation of such a Lie superalgebra, there is a natural definition of the derivative and coderivative, which define the (co)homology groups. We prove that a necessary condition to have a resolution of an irreducible module in terms of Verma modules is complete reducibility of the cohomology groups. Essentially, if it exists, every such a resol…

Centers of universal enveloping algebras of Lie superalgebras in prime characteristic

Let $\ggg=\ggg_\bz+\ggg_\bo$ be a basic classical Lie superalgebra over an algebraically closed field $k$ of characteristic $p>2$, and $G$ be an algebraic supergroup satisfying $\Lie(G)=\ggg$, with the purely even subgroup $G_\ev$ which is a reductive group. The center $\cz:=\cz(\ggg)$ of the universal enveloping algebra of $\ggg$ easily turns out to be a domain. In this paper, we prove that the quotient field of $\cz$ coincides with that of the subalgebra generated by the $G_{\ev}$-invariant ring $\cz^{G_\ev}$ of $\cz$ and the $p$-center $\cz_0$ of $U(\ggg_\bz)$.

Weyl denominator identity for affine Lie superalgebras with non-zero dual Coxeter number

Weyl denominator identity for the affinization of a basic Lie superalgebra with a non-zero Killing form was formulated by Kac and Wakimoto and was proven by them in defect one case. In this paper we prove this identity.

拟约化李超代数的Whittaker范畴与主有限W-超代数

Whittaker categories of quasi-reductive Lie superalgebras and principal finite W-superalgebras

本文研究拟约化李超代数的Whittaker范畴，证明Backelin函子的不可约性，构造主有限W-超代数并建立Skryabin型等价，给出Soergel结构定理的超类比。

等变映射李超代数的Weyl模

Weyl modules for equivariant map Lie superalgebras

本文将非扭情形的Weyl模推广到等变（扭）映射李超代数，定义了整体、局部Weyl模及Weyl函子，证明其泛性质、有限生成性、有限维性及扭模与非扭模的对应关系。

Representations of Lie Superalgebras in Prime Characteristic I

We initiate the representation theory of restricted Lie superalgebras over an algebraically closed field of characteristic p>2. A superalgebra generalization of the celebrated Kac-Weisfeiler Conjecture is formulated, which exhibits a mixture of p-power and 2-power divisibilities of dimensions of modules. We establish the Conjecture for basic classical Lie superalgebras.

关于仿射李超代数及其模的顶点超代数的积分形式

On integral forms for vertex superalgebras associated with affine Lie superalgebras and their modules

本文研究仿射顶点超代数及其模的积分形式，通过构造仿射李超代数通用包络超代数的积分形式，并利用Chevalley基推广Kostant-Garland积分形式理论，进而构造基于仿射李超代数的顶点（算子）超代数及其模的积分形式，并探讨积分形式包含共形向量以及反梯度模中的积分形式。

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.

Characters of (relatively) integrable modules over affine Lie superlagebras

In the paper we consider the problem of computation of characters of relatively integrable irreducible highest weight modules $L$ over finite-dimensional basic Lie superalgebras and over affine Lie superalgebras $\mathfrak{g}$. The problems consists of two parts. First, it is the reduction of the problem to the $\overline{\mathfrak{g}}$-module $F(L)$, where $\overline{\mathfrak{g}}$ is the associated to $L$ integral Lie superalgebra and $F(L)$ is an integrable irreducible highest weight $\overline{\mathfrak{g}}$-module. Second, it is the computation of characters of integrable highest weight m…

Z2 and Klein graded Lie algebras

In this master's thesis, we recall the definitions and basic results for Lie superalgebras. We specify the definition for Klein graded Lie algebras and, motivated by well known results for Lie superalgebras, we prove similar results for Klein graded Lie algebras. More precisely, we state the theorems of Poincare-Birkhoff-Witt and Ado's, as well as Schur's lemma. Moreover, we present two examples of Klein graded algebra.

Classification of finite dimensional modules of singly atypical type over the Lie superalgebras sl(m/n)

We classify the finite dimensional indecomposable sl(m/n)-modules with at least a typical or singly atypical primitive weight. We do this classification not only for weight modules, but also for generalized weight modules. We obtain that such a generalized weight module is simply a module obtained by ``linking'' sub-quotient modules of generalized Kac-modules. By applying our results to sl(m/1), we have in fact completely classified all finite dimensional sl(m/1)-modules.