Bilinear forms on $\mathfrak{sl}_2$-modules and a hypergeometric identity

An analogue of the Kostant criterion for quadratic Lie superalgebras

Let $\mathfrak{r}$ be a finite dimensional complex Lie superalgebra with a non-degenerate super-symmetric invariant bilinear form, let $\mathfrak{p}$ be a finite dimensional complex super vector space with a non-degenerate super-symmetric bilinear form, and let $ν: \mathfrak{r}\rightarrow\mathfrak{osp}(\mathfrak{p})$ be a Lie superalgebra homomorphism. In this paper, we give a necessary and sufficient condition for $\mathfrak{r}\oplus\mathfrak{p}$ to be a quadratic Lie superalgebra. The criterion obtained is an analogue of a constancy condition given by Kostant in the Lie algebra setting. As a…

Quantum alpha-determinants and q-deformed hypergeometric polynomials

The quantum $α$-determinant is defined as a parametric deformation of the quantum determinant. We investigate the cyclic $\mathcal{U}_q(\mathfrak{sl}_2)$-submodules of the quantum matrix algebra $\mathcal{A}_q(\mathrm{Mat}_2)$ generated by the powers of the quantum $α$-determinant. For such a cyclic module, there exists a collection of polynomials which describe the irreducible decomposition of it in the following manner: (i) each polynomial corresponds to a certain irreducible $\mathcal{U}_q(\mathfrak{sl}_2)$-module, (ii) the cyclic module contains an irreducible submodule if the parameter is…

代数$\frak{gl}_3$有限维表示的经典$6j$-符号

Classical $6j$-symbols for finite dimensional representation of the algebra $\mathfrak{gl}_3$

本文推导了代数$\frak{gl}_3$有限维不可约表示的任意$6j$-符号的显式公式，将其表示为代入$\pm 1$的超几何型级数，并推导了选择规则。

The marked Brauer category

We introduce the marked Brauer algebra and the marked Brauer category. These generalize the analogous constructions for the ordinary Brauer algebra to the setting of a homogeneous bilinear form on a $\mathbb{Z}_2$-graded vector space. We classify the simple modules of the marked Brauer algebra over any field of characteristic not two. Under suitable assumptions we show that the marked Brauer algebra is in Schur-Weyl duality with the Lie superalgebra, $\mathfrak{g}$, of linear maps which leave the bilinear form invariant. We also provide a classification of the indecomposable summands of the te…

Cosets of affine vertex algebras inside larger structures

Given a finite-dimensional reductive Lie algebra $\mathfrak{g}$ equipped with a nondegenerate, invariant, symmetric bilinear form $B$, let $V^k(\mathfrak{g},B)$ denote the universal affine vertex algebra associated to $\mathfrak{g}$ and $B$ at level $k$. Let $\mathcal{A}^k$ be a vertex (super)algebra admitting a homomorphism $V^k(\mathfrak{g},B)\rightarrow \mathcal{A}^k$. Under some technical conditions on $\mathcal{A}^k$, we characterize the coset $\text{Com}(V^k(\mathfrak{g},B),\mathcal{A}^k)$ for generic values of $k$. We establish the strong finite generation of this coset in full generali…

Lie algebras constructed with Lie modules and their positively and negatively graded modules

In this paper, we shall give a way to construct a graded Lie algebra $L(\mathfrak{g},ρ,V,{\cal V},B_0)$ from a standard pentad $(\mathfrak{g},ρ,V,{\cal V},B_0)$ which consists of a Lie algebra $\mathfrak{g}$ which has a non-degenerate invariant bilinear form $B_0$ and $\mathfrak{g}$-modules $(ρ, V)$ and ${\cal V}\subset \mathrm {Hom }(V,\mathfrak{k})$ all defined over a field $\mathfrak{k}$. In general, we do not assume that these objects are finite-dimensional. We can embed the objects $\mathfrak{g},ρ,V,{\cal V}$ into $L(\mathfrak{g},ρ,V,{\cal V},B_0)$. Moreover, we construct specific positiv…

Stokes 现象与量子超群 $U_q(\mathfrak{gl}(m|n))$

Stokes phenomenon and quantum supergroup $U_q(\mathfrak{gl}(m|n))$

本文研究与李超代数 $\mathfrak{gl}_{m|n}$ 相关的量子合流超几何超系统的 Stokes 现象，证明其 Stokes 超矩阵满足 Yang-Baxter 方程，从而导出量子超群 $U_q(\mathfrak{gl}(m|n))$。

超几何积分、钩子公式与Whittaker向量

Hypergeometric integrals, hook formulas and Whittaker vectors

本文确定了两类与Young图相关的多维超几何积分之间的比例系数，即加权钩子乘积的倒数，并由此给出gl_n泛包络代数中心在Whittaker向量空间上作用的显式同时特征向量基。

Commuting Traces and Lie Isomorphisms on Generalized Matrix Algebras

Let $\mathcal{G}$ be a generalized matrix algebra over a commutative ring $\mathcal{R}$, ${\mathfrak q}\colon \mathcal{G}\times\mathcal{G}\longrightarrow \mathcal{G}$ be an $\mathcal{R}$-bilinear mapping and ${\mathfrak T}_{\mathfrak q}\colon:\mathcal{G}\longrightarrow \mathcal{G}$ be a trace of $\mathfrak{q}$. We describe the form of ${\mathfrak T}_{\mathfrak q}$ satisfying the condition ${\mathfrak T}_{\mathfrak q}(G)G=G{\mathfrak T}_{\mathfrak q}(G)$ for all $G\in \mathcal{G}$. The question of when ${\mathfrak T}_{\mathfrak q}$ has the proper form is considered. Using the aforementioned tra…

栅栏偏序集、好分次与Frobenius极大抛物子代数

Fence Posets, Good Gradings and Frobenius Maximal Parabolics

本文研究Frobenius极大抛物子代数的主元素伴随作用的特征值重数，证明其在子代数和全矩阵代数上分别关于1/2和0对称且单峰，通过栅栏偏序集与Panyushev约化建立联系，并确定好分次。

Centralizing Traces and Lie Triple Isomorphisms on Triangular Algebras

Let $\mathcal{T}$ be a triangular algebra over a commutative ring $\mathcal{R}$ and $\mathcal{Z(T)}$ be the center of $\mathcal{T}$. Suppose that ${\mathfrak q}\colon \mathcal{T}\times \mathcal{T}\longrightarrow \mathcal{T}$ is an $\mathcal{R}$-bilinear mapping and that ${\mathfrak T}_{\mathfrak q}\colon: \mathcal{T}\longrightarrow \mathcal{T}$ is a trace of $\mathfrak{q}$. We describe the form of ${\mathfrak T}_{\mathfrak q}$ satisfying the condition $[{\mathfrak T}_{\mathfrak q}(T), T]\in \mathcal{Z(T)}$ for all $T\in \mathcal{T}$. The question of when ${\mathfrak T}_{\mathfrak q}$ has the p…

Berezin Kernels and Analysis on Makarevich Spaces

Following ideas of van Dijk and Hille we study the link which exists between maximal degenerate representations and Berezin kernels. We consider the conformal group ${\rm Conf}(V)$ of a simple real Jordan algebra $V$. The maximal degenerate representations $π_s$ ($s\in {\mathbb C}$) we shall study are induced by a character of a maximal parabolic subgroup $\bar P$ of ${\rm Conf}(V)$. These representations $π_s$ can be realized on a space $I_s$ of smooth functions on $V$. There is an invariant bilinear form ${\mathfrak B}_s$ on the space $I_s$. The problem we consider is to diagonalize this bil…

受非退化闭2形式保的受限李超代数双扩展的计算机辅助研究

Computer-aided study of double extensions of restricted Lie superalgebras preserving the non-degenerate closed 2-forms in characteristic 2

本文研究在特征2下，受限李超代数保持非退化闭2形式的双扩展，利用SuperLie工具列举结果，并通过多分级加速计算。

Capelli operators for spherical superharmonics and the Dougall-Ramanujan identity

Let $(V,ω)$ be an orthosympectic $\mathbb Z_2$-graded vector space and let $\mathfrak g:=\mathfrak{gosp}(V,ω)$ denote the Lie superalgebra of similitudes of $(V,ω)$. When the space $\mathscr P(V)$ of superpolynomials on $V$ is \emph{not} a completely reducible $\mathfrak g$-module, we construct a natural basis $D_λ$ of Capelli operators for the algebra of $\mathfrak g$-invariant superpolynomial superdifferential operators on $V$, where the index set $\mathcal P$ is the set of integer partitions of length at most two. We compute the action of the operators $D_λ$ on maximal indecomposable compon…

有限单移位海森堡型群的不可约表示数目

On the number of irreducible representations for finite unipotent Heisenberg-type groups

本文研究了有限单移位海森堡型群的不可约表示数目，通过轨道方法分类，并证明该数目为非负整系数多项式，符合Isaacs猜想。

Screening operators and Parabolic inductions for Affine W-algebras (with an appendix by Shigenori Nakatsuka)

(Affine) $\mathcal{W}$-algebras are a family of vertex algebras defined by the generalized Drinfeld-Sokolov reductions associated with a finite-dimensional reductive Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, a nilpotent element $f$ in $[\mathfrak{g},\mathfrak{g}]$, a good grading $Γ$ and a symmetric invariant bilinear form $κ$ on $\mathfrak{g}$. We introduce free field realizations of $\mathcal{W}$-algebras by using Wakimoto representations of affine Lie algebras, where $\mathcal{W}$-algebras are described as the intersections of kernels of screening operators. We call these Wakimoto free …

偶数和奇数简单G-模的群值动量映射

Group valued moment maps for even and odd simple $G$-modules

研究偶奇简单G-模的动量映射性质，通过条件Hom_g(∧^3V,S^3V)=0，证明其可视为准泊松空间，并构造群值动量映射。

源于格的有理与无理二维共形场论

Rational and non-rational two-dimensional conformal field theories arising from lattices

本文基于实Hilbert空间和正交投影，利用偶格构造了扩展Heisenberg共形网的二维共形网，证明了其酉等价唯一性，并分析了扩展网手征分量的有理性及Wightman场的生成。

仿射旗簇与量子对称对

Affine flag varieties and quantum symmetric pairs

本文通过仿射C型部分旗簇的Schur与Lusztig代数，构造了仿射型量子对称对的几何实现，并建立了相应量子代数的典范基及其正性性质。

有限维Hopf代数的伽马不变量注记

Notes on gamma invariants of finite dimensional Hopf algebras

本文研究有限维非半单Hopf代数的伽马不变量，通过表示理想刻画Green环的传递性，并证明在特定条件下Green环具有Benson意义下的表示环结构，最后应用于Sweedler代数的Drinfeld double和Taft代数。