Hisayosi Matumoto
In this article, we classify the homomorphisms between scalar generalized Verma modules of ${\mathfrak gl}(n, {\mathbb C})$. In fact such homomorphisms are compositions of elementary homomorphisms.
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A homomorphism from the affine Yangian $Y_{\hbar,\varepsilon}(\widehat{\mathfrak{sl}}(n))$ to the affine Yangian $Y_{\hbar,\varepsilon}(\widehat{\mathfrak{sl}}(n+1))$
该研究构造了从n阶仿射Yangian到n+1阶仿射Yangian的同态，并阐明了该同态与作者此前构建的、n阶仿射Yangian到矩形代数泛包络代数同态之间的关联，丰富了仿射Yangian的同态理论。
In this paper, we introduce the Harish-Chandra homomorphism for the quantum superalgebra $\mathrm{U}_q(\mathfrak{g})$ associated with a simple basic Lie superalgebra $\mathfrak{g}$ and give an explicit description of its image. We use it to prove that the center of $\mathrm{U}_q(\mathfrak{g})$ is isomorphic to a subring of the ring $J(\mathfrak{g})$ of exponential super-invariants in the sense of Sergeev and Veselov, establishing a Harish-Chandra type theorem for $\mathrm{U}_q(\mathfrak{g})$. As a byproduct, we obtain a basis of the center of $\mathrm{U}_q(\mathfrak{g})$ with the aid of quasi-…
Let $G_o$ be a semisimple Lie group, let $K_o$ be a maximal compact subgroup of $G_o$ and let $\mathfrak{k}\subset\mathfrak{g}$ denote the complexification of their Lie algebras. Let $G$ be the adjoint group of $\mathfrak{g}$ and let $K$ be the connected Lie subgroup of $G$ with Lie algebra $ad(\mathfrak{k})$. If $U(\mathfrak{g})$ is the universal enveloping algebra of $\mathfrak{g}$ then $U(\mathfrak{g})^K$ will denote the centralizer of $K$ in $U(\mathfrak{g})$. Also let $P:U(\mathfrak{g})\longrightarrow U(\mathfrak{k})\otimes U(\mathfrak{a})$ be the projection map corresponding to the direc…
We construct a class $Θ_{\mathscr{R}}$ of homomorphisms from a Specht module $S_{\mathbb{Z}}^λ$ to a signed permutation module $M_{\mathbb{Z}}(α|β)$ which generalises James's construction of homomorphisms whose codomain is a Young permutation module. We show that any $ϕ\in \operatorname{Hom}_{\mathbb{Z}\mathfrak{S}_{n}}\big(S_{\mathbb{Z}}^λ, M_{\mathbb{Z}}(α|β)\big)$ lies in the $\mathbb{Q}$-span of $Θ_{\text{sstd}}$, a subset of $Θ_{\mathscr{R}}$ corresponding to semistandard $λ$-tableaux of type $(α|β)$. We also study the conditions for which $Θ^{\mathbb{F}}_{\mathrm{sstd}}$ - a subset of $\…
Notes on a homomorphism from the affine Yangian associated with $\hat{\mathfrak{sl}}(n)$ to the affine Yangian associated with $\widehat{\mathfrak{sl}}(n+1)$
本文利用先前构造的affine Yangian同态，构造了从affine Yangian到A型非矩形W-代数泛包络代数的同态，探讨了affine shifted Yangian的新定义，并证明了其与量子仿射代数同态的关联。
The paper mainly considers the center of two-parameter quantum groups $U_{r,s}(\mathfrak{so}_{2n+1})$ via an analogue of the Harish-Chandra homomorphism. In the case when $n$ is odd, the Harish-Chandra homomorphism is not injective in general. When $n$ is even, the Harish-Chandra homomorphism is injective and the center of two-parameter quantum groups $U_{r,s}{(\mathfrak{so}_{2n+1})}$ is described, up to isomorphism.
A Lie algebra $\mathfrak{g}_\mathbb{Q}$ over $\mathbb{Q}$ is said to be $\mathbb{R}$-universal if every homomorphism from $\mathfrak{g}_\mathbb{Q}$ to $\mathfrak{gl}(n,\mathbb{R})$ is conjugate to a homomorphism into $\mathfrak{gl}(n,\mathbb{Q})$ (for every $n$). By using Galois cohomology, we provide a short proof of the known fact that every real semisimple Lie algebra has an $\mathbb{R}$-universal $\mathbb{Q}$-form. We also provide a classification of the $\mathbb{R}$-universal Lie algebras that are semisimple.
Mixed Tensor Products, Capelli Berezinians, and Newton's Formula for $\mathfrak{gl}(m|n)$
本研究将混合张量积与Capelli行列式的相关结果推广到超代数情形，构造了带中心元参数的超代数同态，明确了Gelfand生成元的像，证明了超版本牛顿公式，还给出了膨胀表示为单表示的条件及特定同态的核结构。
Construction of Affine Super Yangian
本文构造带余积结构的仿射超 Yangian，并给出其到仿射 gl 超李代数泛包络代数完备化的赋值同态。
Relating homomorphism spaces between Specht modules of different degrees
针对 Hemmer 提出的不同次数 Specht 模同态空间关联问题，本文证明在奇特征及若干分拆条件满足时，添加 kp^d 前后的两个同态空间同构。
This paper is devoted to an elementary new construction of $1$-singular Gelfand-Tsetlin modules using complex geometry. We introduce a universal ring $\mathcal D_o$ together with the vector space $\mathcal S=\mathcal S(\mathcal D_o)$ with basis $\mathcal B_o = \mathcal B(\mathcal D_o)$ formed from some local distributions such that $\mathcal S$ is a natural $\mathcal D_o$-module. For any homomorphism of rings $\mathcal U(\mathfrak{h}) \to \mathcal D_o$, where $\mathfrak{h}$ is a Lie algebra, it follows that $\mathcal S$ is also an $\mathfrak{h}$-module. We observe that the homomorphism of ring…
Johnson graphs as slices of a hypercube and an algebra homomorphism from the universal Racah algebra into $U(\mathfrak{sl}_2)$
该研究从Johnson图是超立方体切片的视角出发，构造了泛Racah代数到$U(\mathfrak{sl}_2)$的新代数同态，刻画了其核，证明拉回得到的有限维$\Re$-模完全可约且生成元作用为Leonard三元组，还给出了Leonard三元组的具体构造方式。
Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra with associated Yangian $Y_\hbar\mathfrak{g}$ and Yangian double $\mathrm{D}Y_\hbar\mathfrak{g}$. An elementary result of fundamental importance to the theory of Yangians is that, for each $c\in \mathbb{C}$, there is an automorphism $τ_c$ of $Y_\hbar\mathfrak{g}$ corresponding to the translation $t\mapsto t+c$ of the complex plane. Replacing $c$ by a formal parameter $z$ yields the so-called formal shift homomorphism $τ_z$ from $Y_\hbar\mathfrak{g}$ to the polynomial algebra $Y_\hbar\mathfrak{g}[z]$. We prove that $τ_z$ uniquely extends t…
Suppose $μ$ is a partition of $n$ and $λ$ a composition of $n$, and let $S^μ$, $M^λ$ denote the Specht module and permutation module defined by Dipper and James for the Iwahori--Hecke algebra $\mathscr{H}_n$ of the symmetric group $\mathfrak{S}_n$. We give an explicit fast algorithm for expressing a tableau homomorphism $\hatϕ_A:S^μ\to M^λ$ as a linear combination of semistandard homomorphisms. Along the way we provide a utility result related to removing rows from tableaux.
This paper concerns the relation between the quantum toroidal algebras and the affine Yangians of $\mathfrak{sl}_n$, denoted by $\mathcal{U}^{(n)}_{q_1,q_2,q_3}$ and $\mathcal{Y}^{(n)}_{h_1,h_2,h_3}$, respectively. Our motivation arises from the milestone work of Gautam and Toledano Laredo, where a similar relation between the quantum loop algebra $U_q(L \mathfrak{g})$ and the Yangian $Y_h(\mathfrak{g})$ has been established by constructing an isomorphism of $\mathbb{C}[[\hbar]]$-algebras $Φ:\widehat{U}_{\exp(\hbar)}(L\mathfrak{g})\to \widehat{Y}_\hbar(\mathfrak{g})$ (with $\ \widehat{}\ $ sta…
An Example of Homomorphisms from the Guay's affine Yangians to non-rectangular $W$-algebras
本文构造了从Guay仿射Yangian到非矩形$W$-代数泛包络代数的非平凡同态，其中$W$-代数与$\mathfrak{gl}(m+n)$及特定幂零元素相关。
In earlier work, Barchini, Kable, and Zierau constructed a number of conformally invariant systems of differential operators associated to Heisenberg parabolic subalgebras in simple Lie algebras. The construction was systematic, but the existence of such a system was left open in two cases, namely, the $Ω_3$ system for type $A_2$ and type $D_4$. Here, such a system is shown to exist for both cases. The construction of the system may also be interpreted as giving an explicit homomorphism between generalized Verma modules.
Let $\mathfrak{g}$ be a semi-simple Lie algebra with fixed root system, and $U_q(\mathfrak{g})$ the quantization of its universal enveloping algebra. Let $\mathcal{S}$ be a subset of the simple roots of $\mathfrak{g}$. We show that the defining relations for $U_q(\mathfrak{g})$ can be slightly modified in such a way that the resulting algebra $U_q(\mathfrak{g};\mathcal{S})$ allows a homomorphism onto (an extension of) the algebra $\mathrm{Pol}(\mathbb{G}_q/\mathbb{K}_{\mathcal{S},q})$ of functions on the quantum flag manifold $\mathbb{G}_q/\mathbb{K}_{\mathcal{S},q}$ corresponding to $\mathcal…
Affine Super Yangians and Rectangular $W$-superalgebras
该研究构造了仿射超 Yangian 到矩形 $W$-超代数泛包络代数的同态，并给出其像稠密的参数条件。
Let ${\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$ be the double Ringel--Hall algebra of the cyclic quiver $\triangle(n)$ and let $\dot{\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$ be the modified quantum affine algebra of ${\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$. We will construct an integral form $\dot{{\mathfrak D}_{\vartriangle}}(n)$ for $\dot{\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$ such that the natural algebra homomorphism from $\dot{{\mathfrak D}_{\vartriangle}}(n)$ to the integral affine quantum Schur algebra is surjective. Furthermore, we will use Hall algebras to construct…
Exponential modules of $ \mathfrak{osp}(1|2)$
本文研究了由非常值多项式参数化的正交辛李超代数 $\mathfrak{osp}(1|2)$ 的两族非权模，它们源于振子同态与第一 Weyl 代数的指数模，并证明了其单性与同构定理。
Let $Γ$ be a group acting on a scheme $X$ and on a Lie superalgebra $\mathfrak{g}$, both defined over an algebraically closed field of characteristic zero $\Bbbk$. The corresponding equivariant map superalgebra $M(\mathfrak{g}, X)^Γ$ is the Lie superalgebra of equivariant regular maps from $X$ to $\mathfrak{g}$. In this paper we complete the classification of finite-dimensional irreducible $M(\mathfrak{g}, X)^Γ$-modules when $\mathfrak{g}$ is a finite-dimensional simple Lie superalgebra, $X$ is of finite type and $Γ$ is a finite abelian group acting freely on the rational points of $X$, by cla…
In this paper we study properties of a homomorphism $ρ$ from the universal enveloping algebra $U=U(\mathfrak{gl}(n+1))$ to a tensor product of an algebra $\mathcal D'(n)$ of differential operators and $U(\mathfrak{gl}(n))$. We find a formula for the image of the Capelli determinant of $\mathfrak{gl}(n+1)$ under $ρ$, and, in particular, of the images under $ρ$ of the Gelfand generators of the center $Z(\mathfrak{gl}(n+1))$ of $U$. This formula is proven by relating $ρ$ to the corresponding Harish-Chandra isomorphisms, and, alternatively, by using a purely computational approach. Furthermore, we…
A skew group ring of $\mathbb Z/2\mathbb Z$ over $U(\mathfrak{sl}_2)$, Leonard triples and odd graphs
本文利用斜群环构造普遍 Bannai--Ito 代数模，给出 Leonard 三元组作用条件，并建立到奇图 Terwilliger 代数的同态，统一描述所有不可约模上的 Leonard 三元组。
In this note we give defining relations of an $\mathfrak{sl}_{n+1}[t]$-module defined by the fusion product of simple $\mathfrak{sl}_{n+1}$-modules whose highest weights are multiples of a given fundamental weight. From this result we obtain a surjective homomorphism between two fusion products, which can be considered as a current algebra analog of Schur positivity.
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…
On representations of quantum affine $\mathfrak{sl}_2$
本文研究了在任意 $q$ 值下两个任意赋值 $U_q\widehat{\mathfrak{sl}}_2$-模的张量积及其同态等密切相关课题。
In earlier work, Barchini, Kable, and Zierau constructed a number of conformally invariant systems of differential operators associated to Heisenberg parabolic subalgebras in simple Lie algebras. The construction was systematic, but the existence of such a system was left open in several anomalous cases. Here, a conformally invariant system is shown to exist in the most interesting of these remaining cases. The construction may also be interpreted as giving an explicit homomorphism between generalized Verma modules for the Lie algebra of type $D_4$.
We develop the non-commutative polynomial version of the invariant theory for the quantum general linear supergroup ${\rm{ U}}_q(\mathfrak{gl}_{m|n})$. A non-commutative ${\rm{ U}}_q(\mathfrak{gl}_{m|n})$-module superalgebra $\mathcal{P}^{k|l}_{\,r|s}$ is constructed, which is the quantum analogue of the supersymmetric algebra over $\mathbb{C}^{k|l}\otimes \mathbb{C}^{m|n}\oplus \mathbb{C}^{r|s}\otimes (\mathbb{C}^{m|n})^{\ast}$. We analyse the structure of the subalgebra of ${\rm{ U}}_q(\mathfrak{gl}_{m|n})$-invariants in $\mathcal{P}^{k|l}_{\,r|s}$ by using the quantum super analogue of Howe…
Given an algebraic Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, we canonically associate to it a Lie algebra $\mathfrak{g}_{\infty}$ defined over $\mathbb{C}_{\infty}$-the reduction of $\mathbb{C}$ mod infinitely large prime, and show that for a class of Lie algebras $\mathfrak{g}_{\infty}$ is an invariant of the derived category of $\mathfrak{g}$-modules. We give two applications of this construction. First, we show that the bounded derived category of $\mathfrak{g}$-modules determines algebra $\mathfrak{g}$ for a class of Lie algebras. Second, given a semi-simple Lie algebra $\mathfrak{g}$ …
Branching rules for level-zero extremal weight modules from $U_q(\widehat{\mathfrak{sl}}_{n+1})$ to $U_q(\widehat{\mathfrak{sl}}_n)$
本文研究量子仿射代数 $U_q(\widehat{\mathfrak{sl}}_{n+1})$ 的零级极值权模在子代数 $U_q(\widehat{\mathfrak{sl}}_n)$ 下的分支结构，建立了直和分解，并证明在基本权倍数情形下分解为极值权模的直和。
We construct and investigate the structure of the Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^λ$ which give a categrification of quantum generalized Kac-Moody algebras. Let $U_\A(\g)$ be the integral form of the quantum generalized Kac-Moody algebra associated with a Borcherds-Cartan matrix $A=(a_{ij})_{i,j \in I}$ and let $K_0(R)$ be the Grothedieck group of finitely generated projective graded $R$-modules. We prove that there exists an injective algebra homomorphism $Φ: U_\A^-(\g) \to K_0(R)$ and that $Φ$ is an isomorphism if $a_{ii}\ne 0$ for all $i\in I$. Let $B(…
The affine evaluation map is a surjective homomorphism from the quantum toroidal ${\mathfrak {gl}}_n$ algebra ${\mathcal E}'_n(q_1,q_2,q_3)$ to the quantum affine algebra $U'_q\widehat{\mathfrak {gl}}_n$ at level $κ$ completed with respect to the homogeneous grading, where $q_2=q^2$ and $q_3^n=κ^2$. We discuss ${\mathcal E}'_n(q_1,q_2,q_3)$ evaluation modules. We give highest weights of evaluation highest weight modules. We also obtain the decomposition of the evaluation Wakimoto module with respect to a Gelfand-Zeitlin type subalgebra of a completion of ${\mathcal E}'_n(q_1,q_2,q_3)$, which d…
This paper examines the relationship between certain non-commutative analogues of projective 3-space, $\mathbb{P}^3$, and the quantized enveloping algebras $U_q(\mathfrak{sl}_2)$. The relationship is mediated by certain non-commutative graded algebras $S$, one for each $q \in \mathbb{C}^\times$, having a degree-two central element $c$ such that $S[c^{-1}]_0 \cong U_q(\mathfrak{sl}_2)$. The non-commutative analogues of $\mathbb{P}^3$ are the spaces $\operatorname{Proj}_{nc}(S)$. We show how the points, fat points, lines, and quadrics, in $\operatorname{Proj}_{nc}(S)$, and their incidence relati…
A uniform construction of Chevalley normal forms for automorphic Lie algebras on the Riemann sphere
本研究针对SU(2)有限子群相关的自守李代数，统一构造了不变量模生成元与缠结算子，给出单李代数情形下类Chevalley基的生成元，解释了不同对称群自守李代数同构现象，并算出例外李型的结构常数。
For any $n\in \mathbb{Z}_{\geq 2}$, let $\mathfrak{m}_n$ be the subalgebra of $\mathfrak{sp}_{2n}$ spanned by all long negative root vectors $X_{-2ε_i}$, $i=1,\dots,n$. An $\mathfrak{sp}_{2n}$-module $M$ is called a Whittaker module with respect to the Whittaker pair $(\mathfrak{sp}_{2n},\mathfrak{m}_n)$ if the action of $\mathfrak{m}_n$ on $M$ is locally finite, according to a definition of Batra and Mazorchuk. This kind of modules are more general than the classical Whittaker modules defined by Kostant. In this paper, we show that each non-singular block $\mathcal{WH}_{\mathbf{a}}^μ$ with fi…
We begin a study of the representation theory of quantum continuous $\mathfrak{gl}_\infty$, which we denote by $\mathcal E$. This algebra depends on two parameters and is a deformed version of the enveloping algebra of the Lie algebra of difference operators acting on the space of Laurent polynomials in one variable. Fundamental representations of $\mathcal E$ are labeled by a continuous parameter $u\in {\mathbb C}$. The representation theory of $\mathcal E$ has many properties familiar from the representation theory of $\mathfrak{gl}_\infty$: vector representations, Fock modules, semi-infinit…
Representations of Super Yangians with Gelfand-Tsetlin bases
研究超杨代数的表示，通过评价同态将其协变张量模视为超杨代数模，并给出简单商模块的 tame 条件，扩展了纳扎罗夫和塔拉索夫的工作。
We classify and explicitly describe homomorphisms of Verma modules for conformal Galilei algebras $\mathfrak{cga}_\ell(d,{\mathbb C})$ with $d=1$ for any integer value $\ell \in \mathbb{N}$. The homomorphisms are uniquely determined by singular vectors as solutions of certain differential operators of flag type, and identified with specific polynomials arising as coefficients in the expansion of a parametric family of symmetric polynomials into power sum symmetric polynomials.
Let $\mathfrak{g}_m=\mathfrak{sl}(2)\ltimes V(m)$, $m\ge 1$, where $V(m)$ is the irreducible $\mathfrak{sl}(2)$-module of dimension $m+1$ viewed as an abelian Lie algebra. It is known that the isomorphism classes of uniserial $\mathfrak{g}_m$-modules consist of a family, say of type $Z$, containing modules of arbitrary composition length, and some exceptional modules with composition length $\le 4$. Let $V$ and $W$ be two uniserial $\mathfrak{g}_m$-modules of type $Z$. In this paper we obtain the $\mathfrak{sl}(2)$-module decomposition of $\text{soc}(V\otimes W)$ by giving explicitly the highe…
This paper generalizes the results of the paper \cite{mi3} to the case of the general $\mathfrak{sl}_2$ Schubert varieties. We study the homomorphisms between different Schubert varieties, describe their geometry and the group of the line bundles. We also derive some consequences, concerning the infinite-dimensional generalized affine grassmanians.
Let $\mathcal{L}$ be the derivation Lie algebra of ${\mathbb C}[t_1^{\pm 1},t_2^{\pm 1}]$. Given a triangle decomposition $\mathcal{L} =\mathcal{L}^{+}\oplus\mathfrak{h}\oplus\mathcal{L}^{-}$, we define a nonsingular Lie algebra homomorphism $ψ:\mathcal{L}^{+}\rightarrow\mathbb{C}$ and the universal Whittaker $\mathcal{L}$-module $W_ψ$ of type $ψ$. We obtain all Whittaker vectors and submodules of $W_ψ$, and all simple Whittaker $\mathcal{L}$-modules of type $ψ$.
Assume that $\mathbb F$ is an algebraically closed field with characteristic zero. The universal Racah algebra $\Re$ is a unital associative $\mathbb F$-algebra generated by $A,B,C,D$ and the relations state that $[A,B]=[B,C]=[C,A]=2D$ and each of $$ [A,D]+AC-BA, \qquad [B,D]+BA-CB, \qquad [C,D]+CB-AC $$ is central in $\Re$. The universal additive DAHA (double affine Hecke algebra) $\mathfrak H$ of type $(C_1^\vee,C_1)$ is a unital associative $\mathbb F$-algebra generated by $\{t_i\}_{i=0}^3$ and the relations state that \begin{gather*} t_0+t_1+t_2+t_3 = -1, \\ \hbox{$t_i^2$ is central for al…
Irreducibility of the tensor product of Yangian \( \mathrm{Y}(\mathfrak{gl}_{m|n}) \) evaluation modules
本文研究了超Yangian $ \mathrm{Y}(\mathfrak{gl}_{m|n}) $ 评估模块张量积的不可约性条件，基于Gelfand-Tsetlin基的存在性，给出了简单模块的判据。
Derivations on the triplet $W$-algebras with $\mathfrak{sl}_2$-symmetry
本研究细化Tsuchiya-Wood的Frobenius同态，构造带sl₂对称性的三重W-代数上的导子，推广已有导子性质，自然导出其sl₂对称性，还将方法拓展至三重W-超代数并确定其全自同构群。
Let $\mathfrak{g}$ be a simple Lie algebra, and let $W_κ$ be the affine ${W}$-algebra associated to a principal nilpotent element of $\mathfrak{g}$ and level $κ$. We explain a duality between the categories of smooth ${W}$ modules at levels $κ+ κ_c$ and $-κ+ κ_c$, where $κ_c$ is the critical level. Their pairing amounts to a construction of semi-infinite cohomology for the ${W}$-algebra. As an application, we determine all homomorphisms between the Verma modules for ${W}$, verifying a conjecture from the conformal field theory literature of de Vos--van Driel. Along the way, we determine the li…
Let $X$ denote the non-compact globally Hermitian symmetric space of type $DIII$, namely, $\text{SO}(n,\mathbb{H})/\text{U}(n)$. Let $Λ$ be a uniform torsionless lattice in $\text{SO}(n,\mathbb{H})$. In this note we construct certain complex analytic submanifolds in the locally symmetric space $X_Γ:=Γ\backslash \text{SO}(n,\mathbb{H})/\text{U}(n)$ for certain finite index sub lattices $Γ\subset Λ$ and show that their dual cohomology classes in $H^*(X_Γ;\mathbb{C})$ are not in the image of the Matsushima homomorphism $H^*(X_u; \mathbb{C})\to H^*(X_Γ;\mathbb{C})$, where $X_u=\text{SO}(2n)/\text{…
Towards a theta correspondence in families for type II dual pairs
该研究针对p进域上II型一般线性群对偶对，构造了任意交换ℤ[1/p]-代数系数下控制Weil表示中心作用的典范环同态，证明其与标量扩张相容、具有满射性，并联系了Langlands参数模空间与θ对应。
Equivariant K-theory of the space of partial flags
本文通过Drinfeld生成元定义$q=0$仿射量子群代数$\mathfrak{U}_n$，并利用部分旗簇的等变K理论卷积积构造仿射0-Schur代数，证明两者间存在满同态。
Assume that ${\mathbb F}$ is a field with $\operatorname{char}{\mathbb F}\not=2$. The Racah algebra $\Re$ is a unital associative ${\mathbb F}$-algebra defined by generators and relations. The generators are $A$, $B$, $C$, $D$ and the relations assert that $[A,B]=[B,C]=[C,A]=2D$ and each of $[A,D]+AC-BA$, $[B,D]+BA-CB$, $[C,D]+CB-AC$ is central in $\Re$. The Bannai-Ito algebra $\mathfrak{BI}$ is a unital associative ${\mathbb F}$-algebra generated by $X$, $Y$, $Z$ and the relations assert that each of $\{X,Y\}-Z$, $\{Y,Z\}-X$, $\{Z,X\}-Y$ is central in $\mathfrak{BI}$. It was discovered that t…
Odd Verma's Theorem
本文研究了正则对称化Kac-Moody李超代数上Verma超模的性质，通过改变Borel子代数揭示$\mathfrak{gl}(1|1)$结构，并用半砖精确描述主块，改进相伴簇和射影维数结果。
Assume that $\mathbb F$ is an algebraically closed field with characteristic zero. The universal Racah algebra $\Re$ is a unital associative $\mathbb F$-algebra defined by generators and relations. The generators are $A,B, C, D$ and the relations state that $$ [A,B]=[B,C]=[C,A]=2D $$ and each of \begin{gather*} [A,D]+AC-BA, \qquad [B,D]+BA-CB, \qquad [C,D]+CB-AC \end{gather*} is central in $\Re$. The universal additive DAHA (double affine Hecke algebra) $\mathfrak H$ of type $(C_1^\vee,C_1)$ is a unital associative $\mathbb F$-algebra generated by $t_0,t_1,t_0^\vee,t_1^\vee$ and the relations …
Dirac reduction algebra
本文研究正交辛李超代数$\frak{osp}(1|2)$到Weyl-Clifford超代数$W(2n|n)$的同态诱导的Dirac约化代数$Z_n$，给出其局部化的完备表现，并利用该代数生成$n$维平直时空中无质量Dirac方程的所有多项式解。
The Clebsch--Gordan coefficients of $U(\mathfrak{sl}_2)$ and the Terwilliger algebras of Johnson graphs
研究U(sl_2)的Clebsch-Gordan系数与Johnson图Terwilliger代数的联系，通过构造代数同态将张量积模分解为Hahn模，并用二项式系数表达Terwilliger代数维数。
Assume that $\mathbb F$ is an algebraically closed field and $q$ is a nonzero scalar in $\mathbb F$ that is not a root of unity. The universal Askey--Wilson algebra $\triangle_q$ is a unital associative $\mathbb F$-algebra generated by $A,B, C$ and the relations state that each of $$ A+\frac{q BC-q^{-1} CB}{q^2-q^{-2}}, \qquad B+\frac{q CA-q^{-1} AC}{q^2-q^{-2}}, \qquad C+\frac{q AB-q^{-1} BA}{q^2-q^{-2}} $$ is central in $\triangle_q$. The universal DAHA $\mathfrak H_q$ of type $(C_1^\vee,C_1)$ is a unital associative $\mathbb F$-algebra generated by $\{t_i^{\pm 1}\}_{i=0}^3$ and the relation…
On $\mathrm{Ext}^{\bullet}$ between locally analytic generalized Steinberg with applications
本文研究了局部解析广义Steinberg表示之间的Ext群及其应用，推广了Schraen的结果，并定义了更高L-不变量。
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