Blocks of the category $\mathcal{O}^\mathfrak{p}$ in type $E$

On Categories of Admissible $\big(\mathfrak{g},\mathrm{sl}(2)\big)$-Modules

Let $\mathfrak{g}$ be a complex finite-dimensional semisimple Lie algebra and $\mathfrak{k}$ be any $\mathrm{sl}(2)$-subalgebra of $\mathfrak{g}$. In this paper we prove an earlier conjecture by Penkov and Zuckerman claiming that the first derived Zuckerman functor provides an equivalence between a truncation of a thick parabolic category $\mathcal{O}$ for $\mathfrak{g}$ and a truncation of the category of admissible $(\mathfrak{g}, \mathfrak{k})-$modules. This latter truncated category consists of admissible $(\mathfrak{g}, \mathfrak{k})-$modules with sufficiently large minimal $\mathfrak{k}$…

Categorical actions and multiplicities in the Deligne category $\underline{Rep}(GL_t)$

We study the categorical type A action on the Deligne category $\mathcal{D}_t=\underline{Rep}(GL_t)$ (here $t \in \mathbb{C}$) and its "abelian envelope" $\mathcal{V}_t$ constructed in arXiv:1511.07699. For $t \in \mathbb{Z}$, this action categorifies an action of the Lie algebra $\mathfrak{sl}_{\mathbb{Z}}$ on the tensor product of the Fock space $\mathfrak{F}$ with $\mathfrak{F}_t^{\vee}$, its restricted dual "shifted" by $t$, as was suggested by I. Losev. In fact, this action makes the category $\mathcal{V}_t$ the tensor product (in the sense of Losev and Webster, arXiv:1303.1336) of catego…

Automorphisms of finite order, periodic contractions, and Poisson-commutative subalgebras of $\mathcal S(\mathfrak g)$

Let $\mathfrak g$ be a semisimple Lie algebra, $\vartheta\in {\sf Aut}(\mathfrak g)$ a finite order automorphism, and $\mathfrak g_0$ the subalgebra of fixed points of $\vartheta$. Recently, we noticed that using $\vartheta$ one can construct a pencil of compatible Poisson brackets on $\mathcal S(\mathfrak g)$, and thereby a `large' Poisson-commutative subalgebra $\mathcal Z(\mathfrak g,\vartheta)$ of $\mathcal S(\mathfrak g)^{\mathfrak g_0}$. In this article, we study invariant-theoretic properties of $(\mathfrak g,\vartheta)$ that ensure good properties of $\mathcal Z(\mathfrak g,\vartheta)$…

Highest weight categories of $\mathfrak{gl}(\infty)$-modules

We study a category of modules over $\mathfrak{gl}(\infty)$ analogous to category $\mathcal O$. We fix adequate Cartan, Borel and Levi-type subalgebras $\mathfrak h, \mathfrak b$ and $\mathfrak l$ with $\mathfrak l \cong \mathfrak{gl}(\infty)^n$, and define $\mathcal O_{\mathsf{LA}}^{\mathfrak l}{\mathfrak{gl}(\infty)}$ to be the category of $\mathfrak h$-semisimple, $\mathfrak n$-nilpotent modules that satisfy a large annihilator condition as $\mathfrak l$-modules. Our main result is that these are highest weight categories in the sense of Cline, Parshall and Scott. We compute the simple mult…

Positive Systems of Kostant Roots

Let $\mathfrak{g}$ be a simple complex Lie algebra and let $\mathfrak{t} \subset \mathfrak{g}$ be a toral subalgebra of $\mathfrak{g}$. As a $\mathfrak{t}$-module $\mathfrak{g}$ decomposes as \[\mathfrak{g} = \mathfrak{s} \oplus \big(\oplus_{ν\in \mathcal{R}} \mathfrak{g}^ν\big)\] where $\mathfrak{s} \subset \mathfrak{g}$ is the reductive part of a parabolic subalgebra of $\mathfrak{g}$ and $\mathcal{R}$ is the Kostant root system associated to $\mathfrak{t}$. When $\mathfrak{t}$ is a Cartan subalgebra of $\mathfrak{g}$ the decomposition above is nothing but the root decomposition of $\mathfra…

Categorification of Verma modules and indecomposable projective modules in the category $\mathcal I_{\mathfrak{g}}(\mathfrak{sl}_2)$ for $\mathfrak{sl}_2$

We categorify Verma and indecomposable projective modules in the category $\mathcal I_{\mathfrak{g}}(\mathfrak{sl}_2)$ for $\mathfrak{sl}_2$ using a tensor product decomposition theorem of T. J. Enright and work of J. Chuang and R. Rouquier, A. Licata and A. Savage (Hecke algebras, finite general linear groups, and Heisenberg categorification) and M. Khovanov (Heisenberg algebra and a graphical calculus).

环的和-恒等双预层上模范畴的非阿贝尔性

Non-abelianess of the category of modules over a sum-id bipresheaf of rings

本文受 MacPherson 与 Patel 的阿贝尔群双层定义启发，引入小范畴上环的双预层及其模范畴，并通过线性 Grothendieck 构造证明和-恒等双预层上的模范畴可刻画为相应构造上的阿贝尔群双预层范畴，从而证明其非阿贝尔性。

On invariant $1$-dimensional representations of a finite $W$-algebra

Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$ and $G$ be the corresponding simply connected algebraic group. Consider a nilpotent element $e\in \mathfrak{g}$, the corresponding element $χ=(e, \bullet)$ in $\mathfrak{g}^*$, and the coadjoint orbit $\mathbb{O}=Gχ$. We are interested in the set $\mathfrak{J}\mathfrak{d}^1(\mathcal{W})$ of codimension $1$ ideals $J\subset \mathcal{W}$ in a finite $W$-algebra $\mathcal{W}=U(\mathfrak{g}, e)$. We have a natural action of the component group $Γ=Z_G(χ)/Z_G^\circ(χ)$ on $\mathfrak{J}\mathfrak{d}^1(\mathcal{W})$. Denote the set of $Γ$-sta…

The three point gauge algebra $\mathcal V\ltimes \mathfrak{sl}(2, \mathcal R) \oplus\left(Ω_{\mathcal R}/d{\mathcal R}\right)$ and an action on a Fock space

The three point current algebra $\mathfrak{sl}(2,\mathcal R)$ where $\mathcal R=\mathbb C[t,t^{-1},u\,|\,u^2=t^2+4t ]$ and three-point Virasoro algebra both act on a previously constructed Fock space. In this paper we prove that the semi-direct product, i.e. the gauge algebra acts on the Fock space as well.

Tensor representations of $\mathfrak q(\infty)$

We introduce a symmetric monoidal category of modules over the direct limit queer superalgebra $\q (\infty)$. The category can be defined in two equivalent ways with the aid of the large annihilator condition. Tensor products of copies of the natural and the conatural representations are injective objects in this category. We obtain the socle filtrations and formulas for the tensor products of the indecoposable injectives. In addition, it is proven that the category is Koszul self-dual.

Realizations of the three point algebra $\mathfrak{sl}(2, \mathcal R) \oplus\left(Ω_{\mathcal R}/d{\mathcal R}\right)$

We describe the universal central extension of the three point current algebra $\mathfrak{sl}(2,\mathcal R)$ where $\mathcal R=\mathbb C[t,t^{-1},u\,|\,u^2=t^2+4t ]$ and construct realizations of it in terms of sums of partial differential operators.

Local duality for the singularity category of a finite dimensional Gorenstein algebra

A duality theorem for the singularity category of a finite dimensional Gorenstein algebra is proved. It complements a duality on the category of perfect complexes, discovered by Happel. One of its consequences is an analogue of Serre duality, and the existence of Auslander-Reiten triangles for the $\mathfrak{p}$-local and $\mathfrak{p}$-torsion subcategories of the derived category, for each homogeneous prime ideal $\mathfrak{p}$ arising from the action of a commutative ring via Hochschild cohomology.

A Koszul category of representations of finitary Lie algebras

We find for each simple finitary Lie algebra $\mathfrak{g}$ a category $\mathbb{T}_\mathfrak{g}$ of integrable modules in which the tensor product of copies of the natural and conatural modules are injective. The objects in $\mathbb{T}_\mathfrak{g}$ can be defined as the finite length absolute weight modules, where by absolute weight module we mean a module which is a weight module for every splitting Cartan subalgebra of $\mathfrak{g}$. The category $\mathbb{T}_\mathfrak{g}$ is Koszul in the sense that it is antiequivalent to the category of locally unitary finite-dimensional modules over a c…

Some homological properties of category $\mathcal O$ for Lie superalgebras

For classical Lie superalgebras of type I, we provide necessary and sufficient conditions for a Verma supermodule $Δ(λ)$ to be such that every non-zero homomorphism from another Verma supermodule to $Δ(λ)$ is injective. This is applied to describe the socle of the cokernel of an inclusion of Verma supermodules over the periplectic Lie superalgebras $\mathfrak{pe}(n)$ and, furthermore, to reduce the problem of description of $\mathrm{Ext}^1_{\mathcal O}(L(μ),Δ(λ))$ for $\mathfrak{pe}(n)$ to the similar problem for the Lie algebra $\mathfrak{gl}(n)$. Additionally, we study the projective and inj…

Combinatorial and geometric constructions associated with the Kostant cascade

Let $\mathfrak g$ be a complex simple Lie algebra and $\mathfrak b=\mathfrak t\oplus\mathfrak u^+$ a fixed Borel subalgebra. Let $Δ^+$ be the set of positive roots associated with $\mathfrak u^+$ and $\mathcal K\subsetΔ^+$ the Kostant cascade. We elaborate on some constructions related to $\mathcal K$ and applications of $\mathcal K$. This includes the cascade element $x_{\mathcal K}$ in the Cartan subalgebra $\mathfrak t$ and properties of certain objects naturally associated with $\mathcal K$: an abelian ideal of $\mathfrak b$, a nilpotent $G$-orbit in $\mathfrak g$, and an involution of $\m…

Poisson-commutative subalgebras of $S(\mathfrak g)$ associated with involutions

The symmetric algebra $S(\mathfrak g)$ of a reductive Lie algebra $\mathfrak g$ is equipped with the standard Poisson structure, i.e., the Lie-Poisson bracket. Poisson-commutative subalgebras of $S(\mathfrak g)$ attract a great deal of attention, because of their relationship to integrable systems and, more recently, to geometric representation theory. The transcendence degree of a Poisson-commutative subalgebra ${\mathcal C}\subset S(\mathfrak g)$ is bounded by the "magic number" $\boldsymbol{b}(\mathfrak g)$ of $\mathfrak g$. The "argument shift method" of Mishchenko-Fomenko was basically th…

Periodic automorphisms, compatible Poisson brackets, and Gaudin subalgebras

Let $\mathfrak g$ be a finite-dimensional Lie algebra. The symmetric algebra $\mathcal S(\mathfrak g)$ is equipped with the standard Lie-Poisson bracket. In this paper, we elaborate on a surprising observation that one naturally associates the second compatible Poisson bracket on $\mathcal S(\mathfrak g)$ to any finite order automorphism $θ$ of $\mathfrak g$. We study related Poisson-commutative subalgebras $\mathcal C$ of $\mathcal S(\mathfrak g)$ and associated Lie algebra contractions of $\mathfrak g$. To obtain substantial results, we have to assume that $\mathfrak g$ is semisimple. Then w…

W-代数在简单李代数上非有理水平的Kazhdan-Lusztig分类

The Kazhdan-Lusztig category of W-algebras of simply-laced Lie algebras at irrational levels

研究W-代数在简单李代数上非有理水平的Kazhdan-Lusztig分类，通过量子哈密顿约简与affine顶点代数建立braided张量等价。

Nilpotent subspaces and nilpotent orbits

Let $G$ be a semisimple algebraic group with Lie algebra $\mathfrak g$. For a nilpotent $G$-orbit $\mathcal O\subset\mathfrak g$, let $d_\mathcal O$ denote the maximal dimension of a subspace $V\subset \mathfrak g$ that is contained in the closure of $\mathcal O$. In this note, we prove that $d_\mathcal O \le \frac{1}{2}\dim\mathcal O$ and this upper bound is attained if and only if $\mathcal O$ is a Richardson orbit. Furthermore, if $V$ is $B$-stable and $\dim V= \frac{1}{2}\dim\mathcal O$, then $V$ is the nilradical of a polarisation of $\mathcal O$. Every nilpotent orbit closure has a disti…

On Categories $\mathcal{O}$ for Root-Reductive Lie Algebras

Let $\mathfrak{g}$ be a root-reductive Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic $0$ with a splitting Borel subalgebra $\mathfrak{b}$ containing a splitting maximal toral subalgebra $\mathfrak{h}$. We study the category $\bar{\mathcal{O}}$ consisting of all $\mathfrak{h}$-weight $\mathfrak{g}$-modules which are locally $\mathfrak{b}$-finite and have finite-dimensional $\mathfrak{h}$-weight spaces. The focus is on very special Borel subalgebras called the Dynkin Borel subalgebras. This paper serves as an initial passage to the understanding of categories $\ma…

The reducible Specht modules for the Hecke algebra $\mathcal{H}_{\mathbb{C},-1}(\mathfrak{S}_n)$

The reducible Specht modules for the Hecke algebra $\mathcal{H}_{\mathbb{F},q}(\mathfrak{S}_n)$ have been classified except when $q=-1$. We prove one half of a conjecture which we believe classifies the reducible Specht modules when $q=-1$ and $\mathbb{F}$ has characteristic 0.

$\mathcal{U}(\mathfrak{h})$-有限模与权模 I：加权函子、几乎相干族与范畴 $\mathfrak{A}^{\text{irr}}$

$\mathcal{U}(\mathfrak{h})$-finite modules and weight modules I: weighting functors, almost-coherent families and category $\mathfrak{A}^{\text{irr}}$

本文研究 $\mathcal{U}(\mathfrak{h})$-有限 $\mathfrak{g}$-模，利用加权函子与几乎相干族刻画无挠、自由性及不可约权模，并完成 C 型、部分完成 A 型的范畴 $\mathfrak{A}^{\text{irr}}$ 分类。

Toward partial Verma functors of $\mathcal{U}^{[r]}(\mathfrak{g})$ and related results

This note extends the Steinberg Tensor product theorem from the Frobenius kernel $G_{(r)}$ to the deformation $\mathcal{U}^{[r]}(\mathfrak{g})$ of its distribution algebra. As a Corollary we proof some conjectures from \cite{Wes}. Further it describes the graded representation theory of $\mathcal{U}^{[r]}(\mathfrak{g})$ at a generic semisimple p-central character as a twist of the category of graded $G_{(r-1)}$-modules. We conclude by explaining this relation for $SL_2$ explicitely as well as computing the center in this case.

Invariants of the orthosymplectic Lie superalgebra and super Pfaffians

Given a complex orthosymplectic superspace $V$, the orthosymplectic Lie superalgebra $\mathfrak {osp}(V)$ and general linear algebra ${\mathfrak {gl}}_N$ both act naturally on the coordinate super-ring $\mathcal{S}(N)$ of the dual space of $V\otimes{\mathbb C}^N$, and their actions commute. Hence the subalgebra $\mathcal{S}(N)^{\mathfrak {osp}(V)}$ of $\mathfrak {osp}(V)$-invariants in $\mathcal{S}(N)$ has a ${\mathfrak {gl}}_N$-module structure. We introduce the space of super Pfaffians as a simple ${\mathfrak {gl}}_N$-submodule of $\mathcal{S}(N)^{\mathfrak {osp}(V)}$, give an explicit formu…

李代数$\mathfrak g$的horospherical分裂与相关$\mathcal S(\mathfrak g)$的泊松可交换子代数

Horospherical splittings of $\mathfrak g$ and related Poisson commutative subalgebras of $\mathcal S(\mathfrak g)$

本文研究了reductive李代数的horospherical分裂，探讨了相关泊松可交换子代数的性质，并推导了Adler-Kostant-Symes理论的部分结果。

Rationality of the exceptional W-algebras $\mathcal{W}_k(\mathfrak{sp}_4,f_{subreg})$ associated with subregular nilpotent elements of $\mathfrak{sp}_4$

We prove the rationality of the exceptional W-algebras associated with the simple Lie algebra $\mathfrak{sp}_4$ and subregular nilpotent elements, proving a new particular case of a conjecture of Kac-Wakimoto. Moreover, we describe the simple $\mathcal{W}_k(\mathfrak{sp}_4,f_{subreg})$-modules and compute their characters. We also explicit the nontrivial action of the component group on the set of these simple modules.

Symmetric structure for the endomorphism algebra of projective-injective module in parabolic category

We show that for any singular dominant integral weight $λ$ of a complex semisimple Lie algebra $\mathfrak{g}$, the endomorphism algebra $B$ of any projective-injective module of the parabolic BGG category $\mathcal{O}_λ^{\mathfrak{p}}$ is a symmetric algebra (as conjectured by Khovanov) extending the results of Mazorchuk and Stroppel for the regular dominant integral weight. Moreover, the endomorphism algebra $B$ is equipped with a homogeneous (non-degenerate) symmetrizing form. In the appendix, there is a short proof due to K. Coulembier and V. Mazorchuk showing that the endomorphism algebra …

Takiff 代数的抛物诱导与 Chari 范畴 $\widetilde{\mathcal{O}}$ 中的特征标公式

Parabolic induction for Takiff algebras and character formulas in Chari's category $\widetilde{\mathcal{O}}$

本文引入并分类 Takiff 代数的抛物子代数，建立抛物诱导模不可约性判据，并推导 Chari 范畴中 Harish-Chandra 模的特征标公式。

李超代数的范畴 $\mathcal{O}$

Category $\mathcal{O}$ for Lie superalgebras

本文为任意拟约化李超代数定义了统一的范畴 $\mathcal{O}$，证明其在主抛物分解下是标准分层的，并建立有限生成范畴上同调及模复杂度的有限性上界，为支撑簇理论奠定基础。

Category O for Takiff Lie algebras

We study category $\mathcal{O}$ for Takiff Lie algebras $\mathfrak{g} \otimes \mathbb{C}[ε]/(ε^2)$ where $\mathfrak{g}$ is the Lie algebra of a reductive algebraic group over $\mathbb{C}$. We decompose this category as a direct sum of certain subcategories and use an analogue of parabolic induction functors and twisting functors for BGG category $\mathcal{O}$ to prove equivalences between these subcategories. We then use these equivalences to compute the composition multiplicities of the simple modules in the Verma modules in terms of composition multiplicities in the BGG category $\mathcal{O}…

Cohomology of $\mathfrak {aff}(1)$ and $\mathfrak {aff}(1|1)$ acting on the space of $n$-ary differential operators on the superspace $\mathbb{R}^{1|1}$

We consider the $μ$-densities spaces $\mathcal{F}_μ$ with $μ\in\mathbb{R}$, we compute the space $\mathrm{H}^1_\mathrm{diff}(\mathfrak{aff}(1),\mathrm{D}_{λ,μ})$ where $λ=(λ_1,\dots,λ_n)\in\mathbb{R}^n$ and $\mathrm{D}_{λ,μ}$ is the space of $n$-ary differential operators from $\mathcal{F}_{λ_1}\otimes\cdots\otimes\mathcal{F}_{λ_n}$ to $\mathcal{F}_μ$. We also compute the super analog space $\mathrm{H}^1_\mathrm{diff}(\mathfrak{aff}(1|1),\mathfrak{D}_{λ,μ})$.

Tensor Triangular Geometry for Classical Lie Superalgebras

Tensor triangular geometry as introduced by Balmer is a powerful idea which can be used to extract the ambient geometry from a given tensor triangulated category. In this paper we provide a general setting for a compactly generated tensor triangulated category which enables one to classify thick tensor ideals and the Balmer spectrum. For a classical Lie superalgebra ${\mathfrak g}={\mathfrak g}_{\bar{0}}\oplus {\mathfrak g}_{\bar{1}}$, we construct a Zariski space from a detecting subalgebra of ${\mathfrak g}$ and demonstrate that this topological space governs the tensor triangular geometry f…

在periplectic李超代数$\mathfrak{pe}(2)$的category $\mathcal{O}$中的Koszul性

Koszulity in the category $\mathcal{O}$ of the periplectic Lie Superalgebra $\mathfrak{pe}(2)$

本文研究了periplectic李超代数$\mathfrak{pe}(2)$的category $\mathcal{O}$中哪些块是Koszul的，通过计算投射模的自同态代数并证明其诱导性，确定了剩余块的Koszul性。

Schur-Weyl duality for Deligne categories II: the limit case

This paper is a continuation of a previous paper of the author, which gave an analogue to the classical Schur-Weyl duality in the setting of Deligne categories. Given a finite-dimensional unital vector space $V$ (a vector space $V$ with a chosen non-zero vector $\mathbf{1}$), we constructed a complex tensor power of $V$: an $Ind$-object of the Deligne category $\underline{Rep}(S_t)$ which is a Harish-Chandra module for the pair $(\mathfrak{gl}(V), \bar{\mathfrak{P}}_{\mathbf{1}})$, where $\bar{\mathfrak{P}}_{\mathbf{1}} \subset GL(V)$ is the mirabolic subgroup preserving the vector $\mathbf{1}…

BGG 范畴 $\mathcal{O}$ 的 Dirac 上同调

Dirac cohomology for the BGG category $\mathcal{O}$

本文研究有限维复半单李代数范畴 $\mathcal{O}$ 中模的 Dirac 上同调，证明 Vogan 猜想，揭示其与幂零李代数上同调的联系，并验证高阶 Dirac 上同调的同调性质。

Non-negative integral level affine Lie algebra tensor categories and their associativity isomorphisms

For a finite-dimensional simple Lie algebra $\mathfrak{g}$, we use the vertex tensor category theory of Huang and Lepowsky to identify the category of standard modules for the affine Lie algebra $\hat{\mathfrak{g}}$ at a fixed level $\ell\in\mathbb{N}$ with a certain tensor category of finite-dimensional $\mathfrak{g}$-modules. More precisely, the category of level $\ell$ standard $\hat{\mathfrak{g}}$-modules is the module category for the simple vertex operator algebra $L_{\hat{\mathfrak{g}}}(\ell,0)$, and as is well known, this category is equivalent as an abelian category to $\mathbf{D}(\ma…

Projective modules over classical Lie algebras of infinite rank in the parabolic category

We study the truncation functors and show the existence of projective cover of each irreducible module in parabolic BGG category $\mathcal O$ over infinite rank Lie algebra of types $\mathfrak{a,b,c,d}$. Moreover, $\mathcal O$ is a Koszul category. As a consequence, the corresponding parabolic BGG category $\overline{\mathcal O}$ over infinite rank Lie superalgebra of types $\mathfrak{a,b,c,d}$ through the super duality is also a Koszul category.

On commuting varieties of nilradicals of Borel subalgebras of reductive Lie algebras

Let $G$ be a connected reductive algebraic group defined over an algebraically closed field $\mathbbm k$ of characteristic zero. We consider the commuting variety $\mathcal C(\mathfrak u)$ of the nilradical $\mathfrak u$ of the Lie algebra $\mathfrak b$ of a Borel subgroup $B$ of $G$. In case $B$ acts on $\mathfrak u$ with only a finite number of orbits, we verify that $\mathcal C(\mathfrak u)$ is equidimensional and that the irreducible components are in correspondence with the {\em distinguished} $B$-orbits in $\mathfrak u$. We observe that in general $\mathcal C(\mathfrak u)$ is not equidim…

Abelian quotients arising from extriangulated categories via morphism categories

We investigate abelian quotients arising from extriangulated categories via morphism categories, which is a unified treatment for both exact categories and triangulated categories. Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with enough projectives $\mathcal{P}$ and $\mathcal{M}$ be a full subcategory of $\mathcal{C}$ containing $\mathcal{P}$. We show that certain quotient category of $\mathfrak{s}\textup{-def}(\mathcal{M})$, the category of $\mathfrak{s}$-deflations $f:M_{1}\rightarrow M_2$ with $M_1,M_2\in\mathcal{M}$, is abelian. Our main theorem has two applic…

不可约 $\mathcal{U}(\mathfrak{g})^{G'}$-模的构造与离散可分解限制

Construction of Irreducible $\mathcal{U}(\mathfrak{g})^{G'}$-Modules and Discretely Decomposable Restrictions

本文在约化李代数分支问题中研究 $\mathcal{U}(\mathfrak{g})^{G'}$-模的不可约性，利用 Zuckerman 派生函子构造有限维不可约模，并在拟交换且良好范围条件下给出广义 Verma 模、上同调诱导模和离散序列表示的不可约性判据及相应分支律。

Stable Hilbert series of $\mathcal S(\mathfrak g)^K$ for classical groups

Given a classical symmetric pair, $(G,K)$, with $\mathfrak g = Lie(G)$, we provide descriptions of the Hilbert series of the algebra of $K$-invariant vectors in the associated graded algebra of $\mathcal U(\mathfrak g)$ viewed as a $K$-representation under restriction of the adjoint representation. The description illuminates a certain stable behavior of the Hilbert series, which is investigated in a case-by-case basis. We note that the stable Hilbert series of one symmetric pair often coincides with others. Also, for the case of the real form $U(p,q)$ we derive a closed expression for the Hil…

Quantum $α$-determinant cyclic modules of $\mathcal{U}_q(\mathfrak{gl}_n)$

As a particular one parameter deformation of the quantum determinant, we introduce a quantum $α$-determinant and study the $\mathcal{U}_q(\mathfrak{gl}_n)$-cyclic module generated by it: We show that the multiplicity of each irreducible representation in this cyclic module is determined by a certain polynomial called the $q$-content discriminant. A part of the present result is a quantum counterpart for the result of Matsumoto and Wakayama (2005), however, a new distinguished feature arises in our situation.

Categories O for Dynkin Borel Subalgebras of Root-Reductive Lie Algebras

The purpose of my Ph.D. research is to define and study an analogue of the classical Bernstein-Gelfand-Gelfand (BGG) category $\mathcal{O}$ for the Lie algebra $\mathfrak{g}$, where $\mathfrak{g}$ is one of the finitary, infinite-dimensional Lie algebras $\mathfrak{gl}_\infty(\mathbb{K})$, $\mathfrak{sl}_\infty(\mathbb{K})$, $\mathfrak{so}_\infty(\mathbb{K})$, and $\mathfrak{sp}_\infty(\mathbb{K})$. Here, $\mathbb{K}$ is an algebraically closed field of characteristic $0$. We call these categories "extended categories $\mathcal{O}$" and use the notation $\bar{\mathcal{O}}$. While the categorie…

李代数背景下有限维 $\mathfrak{sl}_3$-模的无限秩模范畴的组合学

Combinatorics of infinite rank module categories over finite dimensional $\mathfrak{sl}_3$-modules in Lie-algebraic context

本文研究有限维 $\mathfrak{sl}_3$-模幺半范畴作用于任意单模时产生的传递模范畴，确定其组合结构，并得到一族八个图作为经典无限 Dynkin 图的 $\mathfrak{sl}_3$-推广。

The Dunkl-Cherednik Deformation of a Howe duality

We consider the deformed versions of the classical Howe dual pairs $(O(r),\mathfrak{s}\mathfrak{l}(2))$ and $(O(r),\mathfrak{s}\mathfrak{p}\mathfrak{o}(2|2))$ in the context of a rational Cherednik algebra $H_c=H_c(W,\mathfrak{h})$ associated to a finite Coxeter group $W$ at the parameters $c$ and $t=1$. For the first pair, we compute the centraliser of the well-known copy of $\mathfrak{s}\cong\mathfrak{s}\mathfrak{l}(2)$ inside $H_c$. For the second pair, we show that the classical copy of $\mathfrak{g}\cong\mathfrak{s}\mathfrak{p}\mathfrak{o}(2|2)$ inside the Weyl-Clifford algebra $\mathcal{…

Affine category O, Koszul duality and Zuckerman functors

The parabolic category $\mathcal{O}$ for affine ${\mathfrak{gl}}_N$ at level $-N-e$ admits a structure of a categorical representation of $\widetilde{\mathfrak{sl}}_e$ with respect to some endofunctors $E$ and $F$. This category contains a smaller category $\mathbf{A}$ that categorifies the higher level Fock space. We prove that the functors $E$ and $F$ in the category $\mathbf{A}$ are Koszul dual to Zuckerman functors. The key point of the proof is to show that the functor $F$ for the category $\mathbf{A}$ at level $-N-e$ can be decomposed in terms of the components of the functor $F$ for the…

Tensor Triangular Geometry for Quantum Groups

Let $\mathfrak g$ be a complex simple Lie algebra and let $U_ζ({\mathfrak g})$ be the corresponding Lusztig ${\mathbb Z}[q,q^{-1}]$-form of the quantized enveloping algebra specialized to an $\ell$th root of unity. Moreover, let $\mod(U_ζ({\mathfrak g}))$ be the braided monoidal category of finite-dimensional modules for $U_ζ({\mathfrak g})$. In this paper we classify the thick tensor ideals of $\mod(U_ζ({\mathfrak g}))$ and compute the prime spectrum of the stable module category associated to $\text{mod}(U_ζ({\mathfrak g}))$ as defined by Balmer.

Lie wedges of endomorphism semigroups of standard subspaces in admissible Lie algebras

Let $\mathfrak{g}$ be a real finite-dimensional Lie algebra containing pointed generating invariant closed convex cones. We determine those derivations $D$ of $\mathfrak{g}$ which induce a 3-grading of the form $\mathfrak{g} = \mathfrak{g}_{-1} \oplus \mathfrak{g}_0 \oplus \mathfrak{g}_1$ on $\mathfrak{g}$ such that the $(\pm 1)$-eigenspaces $\mathfrak{g}_{\pm 1}$ of $D$ are generated by the intersections with generating cones of the form $W = \mathcal{O}_f^*$, where $\mathcal{O}_f$ is the coadjoint orbit of a linear functional $f \in \mathfrak{z}(\mathfrak{g})^*$ and $\mathcal{O}_f^*$ is the …

Large annihilator category O for sl_{\infty}, o_{\infty}, sp_{\infty}

We construct a new analogue of the BGG category $\mathcal O$ for the infinite-dimensional Lie algebras $\fg=\mathfrak{sl}(\infty),\mathfrak{o}(\infty), \mathfrak{sp}(\infty)$. A main difference with the categories studied in \cite{Nam} and \cite{CP} is that all objects of our category satisfy the large annihilator condition introduced in \cite{DPS}. Despite the fact that the splitting Borel subalgebras $\fb$ of $\fg$ are not conjugate, one can eliminate the dependency on the choice of $\fb$ and introduce a universal highest weight category $\mathcal {OLA}$ of $\fg$-modules, the letters $\mathc…

Cartan subalgebras for restrictions of $\mathfrak{g}$-modules

In this paper, we deal with the $\mathcal{U}(\mathfrak{g})$-action on a $\mathfrak{g}$-module on which a larger algebra $\mathcal{A}$ acts irreducibly. Under a mild condition, we will show that the support of the $\mathcal{Z}(\mathfrak{g})$-action is a union of affine subspaces in the dual of a Cartan subalgebra modulo the Weyl group action. As a consequence, we propose a definition of a Cartan subalgebra for such a $\mathfrak{g}$-module. The support of the $\mathcal{Z}(\mathfrak{g})$-module is an algebraic counterpart of the support of the measure in the irreducible decomposition of a unitary…

Orbifolds and cosets of minimal $\mathcal{W}$-algebras

Let $\mathfrak{g}$ be a simple, finite-dimensional Lie (super)algebra equipped with an embedding of $\mathfrak{s} \mathfrak{l}_2$ inducing the minimal gradation on $\mathfrak{g}$. The corresponding minimal $\mathcal{W}$-algebra $\mathcal{W}^k(\mathfrak{g}, e_{-θ})$ introduced by Kac and Wakimoto has strong generators in weights $1,2,3/2$, and all operator product expansions are known explicitly. The weight one subspace generates an affine vertex (super)algebra $V^{k'}(\mathfrak{g}^{\natural})$ where $\mathfrak{g}^{\natural} \subset \mathfrak{g}$ denotes the centralizer of $\mathfrak{s} \mathfr…

Corwin-Greenleaf multiplicity function for compact extensions of the Heisenberg group

Let $\mathbb{H}_n$ be the $(2n+1)$-dimensional Heisenberg group and $K$ a closed subgroup of $U(n)$ acting on $\mathbb{H}_n$ by automorphisms such that $(K,\mathbb{H}_n)$ is a Gelfand pair. Let $G=K\ltimes\mathbb{H}_n$ be the semidirect product of $K$ and $\mathbb{H}_n$. Let $\mathfrak{g}\supset\mathfrak{k}$ be the respective Lie algebras of $G$ and $K$, and $\operatorname{pr}: \mathfrak{g}^{*}\longrightarrow\mathfrak{k}^{*}$ the natural projection. For coadjoint orbits $\mathcal{O}^{G}\subset\mathfrak{g}^{*}$ and $\mathcal{O}^{K}\subset\mathfrak{k}^{*}$, we denote by $n\big(\mathcal{O}^{G},\m…

有限$W$-代数平移函子的幺半范畴视角

A monoidal category viewpoint for translation functors for finite $W$-algebras

本研究从幺半范畴视角重新解释有限$W$-代数的Goodwin平移函子，通过Skryabin等价构造模作用，并证明分阶段Skryabin等价是$U(\mathfrak{g})$-模范畴的等价，深化了相关代数结构的范畴论理解。

斜范畴代数和环化有限景上的模

Skew category algebras and modules on ringed finite sites

本研究针对小范畴上的环化景及其模范畴，借助Grothendieck与Verdier的有限拓扑斯分类，证明有限景上的模范畴等价于由唯一确定的满子范畴及环层限制所定义的斜范畴代数的模范畴。