A. A. Gerasimov, D. R. Lebedev, S. V. Oblezin
The orthosymplectic super Lie algebra $\mathfrak{osp}(1|\,2\ell)$ is the closest analog of standard Lie algebras in the world of super Lie algebras. We demonstrate that the corresponding $\mathfrak{osp}(1|\,2\ell)$-Toda chain turns out to be an instance of a $BC_\ell$-Toda chain. The underlying reason for this relation is discussed.
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We prove that the finite $\mathcal{W}$-algebra associated to $\mathfrak{osp}_{1|2n}$ and its principal nilpotent element is isomorphic to Gorelik's ghost center of $\mathfrak{osp}_{1|2n}$, which proves an analog of Kostant's theorem for $\mathfrak{osp}_{1|2n}$.
Diagonal reduction algebra for $\mathfrak{osp}(1|2)$
本文研究 $\mathfrak{osp}(1|2)$ 对角约化代数，将其刻画为带结合菱形积的双陪集空间，并给出完整生成元关系表示、PBW 基、类 Casimir 元素及自同构子群。
We study a particular class of infinite-dimensional representations of $\mathfrak{osp}(1|2n)$. These representations $L_n(p)$ are characterized by a positive integer $p$, and are the lowest component in the $p$-fold tensor product of the metaplectic representation of $\mathfrak{osp}(1|2n)$. We construct a new polynomial basis for $L_n(p)$ arising from the embedding $\mathfrak{osp}(1|2np) \supset \mathfrak{osp}(1|2n)$. The basis vectors of $L_n(p)$ are labelled by semi-standard Young tableaux, and are expressed as Clifford algebra valued polynomials with integer coefficients in $np$ variables. …
The Bannai-Ito algebra $BI(n)$ is viewed as the centralizer of the action of $\mathfrak{osp}(1|2)$ in the $n$-fold tensor product of the universal algebra of this Lie superalgebra. The generators of this centralizer are constructed with the help of the universal $R$-matrix of $\mathfrak{osp}(1|2)$. The specific structure of the $\mathfrak{osp}(1|2)$ embeddings to which the centralizing elements are attached as Casimir elements is explained. With the generators defined, the structure relations of $BI(n)$ are derived from those of $BI(3)$ by repeated action of the coproduct and using properties …
Unitary Branching for $\mathfrak{sl}(m\vert n), \mathfrak{osp}(m\vert 2n)$ and $F(4)$
本文通过扩展Dirac方法并利用修正的Dirac不等式与超Freudenthal递归，确定了基本经典李超代数在偶数子代数限制下的酉分支，给出了分支重数的闭式公式及酉性刻画。
We study the effects of the branching $\mathfrak{osp}(1|2n)\supset \mathfrak{gl}(n)$ on a particular class of simple infinite-dimensional $\mathfrak{osp}(1|2n)$-modules $L(p)$ characterized by a positive integer $p$. In the first part we use combinatorial methods such as Young tableaux and Young subgroups to construct a new basis for $L(p)$ that respects this branching and we express the basis elements explicitly in two distinct ways. First as monomials of negative root vectors of $\mathfrak{gl}(n)$ acting on the $\mathfrak{gl}(n)$-highest weight vectors in $L(p)$ and then as polynomials in th…
The paper is devoted to a computation of the Lie superalgebras of holomorphic vector fields on isotropic flag supermanifolds of maximal type corresponding to the Lie superalgebras $\mathfrak{osp}_{2m|2n}(\mathbb C)$ and $π\mathfrak{sp}_{n}(\mathbb C)$. The result is that under some restrictions on the flag type any holomorphic vector field is fundamental with respect to the natural action of the Lie superalgebras $\mathfrak{osp}_{2m|2n}(\mathbb C)$ or $π\mathfrak{sp}_{n}(\mathbb C)$.
We classify deformations of $\mathfrak{osp}(2|2)-$module structure on the spaces of symbols $\mathfrak{S}_d^2$ of differential operators acting on the space of weighted densities $\mathfrak{F}_λ^{2}$.
We introduce a subalgebra $\overline F$ of the Clifford vertex superalgebra ($bc$ system) which is completely reducible as a $L^{Vir} (-2,0)$-module, $C_2$-cofinite, but it is not conformal and it is not isomorphic to the symplectic fermion algebra $\mathcal{SF}(1)$. We show that $\mathcal{SF}(1)$ and $\overline{F}$ are in an interesting duality, since $\overline{F}$ can be equipped with the structure of a $\mathcal{SF}(1)$-module and vice versa. Using the decomposition of $\overline F$ and a free-field realization from arXiv:1711.11342, we decompose $L_k(\mathfrak{osp}(1\vert 2))$ at the crit…
The purpose of this work is to illustrate in a family of interesting examples how to study the representation theory of vertex operator superalgebras by combining the theory of vertex algebra extensions and modular forms. Let $L_k\left(\mathfrak{osp}(1 | 2)\right)$ be the simple affine vertex operator superalgebra of $\mathfrak{osp}(1|2)$ at an admissible level $k$. We use a Jacobi form decomposition to see that this is a vertex operator superalgebra extension of $L_k(\mathfrak{sl}_2)\otimes \text{Vir}(p, (p+p')/2)$ where $k+3/2=p/(2p')$ and $\text{Vir}(u, v)$ denotes the regular Virasoro vert…
The character formula of any finite dimensional irreducible module for Lie superalgebra $\mathfrak{osp}(3|2m)$ is obtained in terms of characters of generalized Verma modules.
We classify irreducible super unitary representations of an orthosymplectic Lie superalgebra $ \mathfrak{osp}(M/N; \mathbb{R} ) $ using the notion of super dual pairs in $ \mathfrak{osp}(M/N) $.
We compute the first cohomology spaces $H^1(\mathfrak{osp}(1|2);\mathfrak{D}_{λ,μ})$ ($λ, μ\in\mathbb{R}$) of the Lie superalgebra $\mathfrak{osp}(1|2)$ with coefficients in the superspace $\mathfrak{D}_{λ,μ}$ of linear differential operators acting on weighted densities on the supercircle $S^{1|1}$. The structure of these spaces was conjectured in \cite{gmo}. In fact, we prove here that the situation is a little bit more complicated. (To appear in LMP.)
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…
We compute the Lie superalgebras of holomorphic vector fields on isotropic flag supermanifolds of maximal type corresponding to the Lie superalgebra $\mathfrak{osp}_{2m-1|2n}(\mathbb C)$.
We compute the first differential cohomology of the orthosymplectic Lie superalgebra $\mathfrak{osp}(2|2)$ with coefficients in the superspace of linear differential operators acting on the space of weighted densities on the (1,\,2)-dimensional real superspace. We also compute the same, but $\mathfrak{osp}(1|2)$-relative, cohomology. We explicitly give 1-cocycles spanning these cohomology. This work is a simplest generalization of a result by Basdouri and Ben Ammar [Cohomology of $\frak {osp}(1|2)$ with coefficients in $\frak{D}_{λ,μ}$.
The character formula of any finite dimensional irreducible module $L_λ$ for Lie superalgebra $\mathfrak{osp}(n|2)$ is computed. As a by-product, the decomposition of tensor module $L_λ\otimes \mathbb{C}^{n|2}$, where $\mathbb{C}^{n|2}$ is the natural representation, is obtained.
Ghost center and representations of the diagonal reduction algebra of $\mathfrak{osp}(1|2)$
本文研究 $\mathfrak{osp}(1|2)$ 的对角约化超代数 $A$，通过 Harish–Chandra 同态、Verma 模与 Shapovalov 型刻画其 ghost center，并分类有限维不可约表示及给出无限维张量积分解。
We determine the Verma multiplicities of standard filtrations of projective modules for integral atypical blocks in the BGG category $\mathcal{O}$ for the orthosymplectic Lie superalgebras $\mathfrak{osp}(3|4)$ by way of translation functors. We then explicitly determine the composition factor multiplicities of Verma modules using BGG reciprocity.
Relative modular categories from $\mathfrak{osp}(2 \vert 2n)$
本文基于$\boldsymbol{\frak{osp}(2|2n)}$的展开量子群加权表示理论，构造了相对模范畴。
The complexity of a module is the rate of growth of the minimal projective resolution of the module while the $z$-complexity is the rate of growth of the number of indecomposable summands at each step in the resolution. Let $\mathfrak{g}=\mathfrak{osp}(k|2)$ ($k>2$) be the type II orthosymplectic Lie superalgebra of types $B$ or $D$. In this paper, we compute the complexity and the $z$-complexity of the simple finite-dimensional $\mathfrak{g}$-supermodules. We then give geometric interpretations using support and associated varieties for these complexities.
Let $\bar{G}$ be the simple algebraic supergroup $\mathrm{SL}(m|n)$ or $\mathrm{OSp}(m|2n)$ over $\mathbb{C}$. Let $\mathfrak{g}=\mathrm{Lie}(\bar{G})=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ and let $G=\bar{G}(\mathbb{C})$ where $\mathbb{C}$ is considered as a superalgebra concentrated in even degree. Suppose $e\in\mathfrak{g}_{\bar{0}}$ is nilpotent. We describe the centralizer $\mathfrak{g}^{e}$ of $e$ in $\mathfrak{g}$ and its centre $\mathfrak{z}(\mathfrak{g}^{e})$. In particular, we give bases for $\mathfrak{g}^{e}$, $\mathfrak{z}(\mathfrak{g}^{e})$ and $\left(\mathfrak{z}(\ma…
Odd nilpotent element and $\mathfrak{osp}(1|2)$-subalgebra in $\mathfrak{gl}(m|n)$
本文研究$\mathfrak{gl}(m|n)$中奇幂零元嵌入$\mathfrak{osp}(1|2)$-子代数的条件，通过定义超Jordan矩阵并证明充要条件：该元位于仅含奇数阶超Jordan块的超Jordan矩阵轨道中。
The Weyl groupoids of $\mathfrak{sl}(m|n)$ and $\mathfrak{osp}(r|2n)$
本文为 Heckenberger 和 Schneider 的 Cartan 图与 Weyl 群胚给出方便表述，构造正则可对称化反变李超代数的 Cartan 图，并用分拆显式描述 sl(m|n) 与 osp(r|2n) 的 Cartan 图及其 Weyl 群胚生成元关系。
The quantum supergroup ${\rm{U}}_q({\mathfrak {osp}}(1|2n))$ admits a finite dimensional spinor representation, which does not have a classical limit. We construct a realisation of this representation on the Fock space of $q$-fermions. We also generalise the construction to the infinite dimensional spinor representations of ${\rm{U}}_q({\mathfrak {osp}}(2m+1|2n))$ for $m, n\ge 1$ by using both quantum bosons and fermions. This leads to a new realisation different from those obtained before by other researchers.
The Kazhdan-Lusztig category of $\mathfrak{osp}_{1|2n}$ at irrational levels
本文证明了 $\mathfrak{osp}_{1|2n}$ 在无理水平下的 Kazhdan-Lusztig 对应，建立了其仿射顶点算子超代数范畴与量子群范畴及 $\mathfrak{so}_{2n+1}$ 对应范畴的辫子张量等价，并构造了新的简单共形顶点（超）代数族。
The minimal representation of a semisimple Lie group is a 'small' infinite-dimensional irreducible unitary representation. It is thought to correspond to the minimal nilpotent coadjoint orbit in Kirillov's orbit philosophy. The Segal-Bargmann transform is an intertwining integral transformation between two different models of the minimal representation for Hermitian Lie groups of tube type. In this paper we construct a Fock model for the minimal representation of the orthosymplectic Lie superalgebra $\mathfrak{osp}(m,2|2n)$. We also construct an integral transform which intertwines the Schrödi…
The main goal of our paper is to establish a connection between the Weyl modules of the current Lie superalgebras (twisted and untwisted) attached to $\mathfrak{osp}(1,2)$ and the nonsymmetric Macdonald polynomials of types $A_2^{(2)}$ and $A_2^{(2)\dagger}$. We compute the dimensions and construct bases of the Weyl modules. We also derive explicit formulas for the t=0 and t=\infty specializations of the nonsymmetric Macdonald polynomials. We show that the specializations can be described in terms of the Lie superalgebras action on the Weyl modules.
Howe duality for the dual pair $\left(\text{SpO}(2n|1)\,, \mathfrak{osp}(2|2)\right)$
本文研究了SpO(2n|1)和osp(2|2)对偶性下的表示分解，并给出了最高权和联合最高权向量的显式描述。
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。
Admissible level $\mathfrak{osp}(1|2)$ minimal models and their relaxed highest weight modules
本文结合两类自由场实现、屏蔽算子与 Jack 基对称函数理论，计算 Zhu 代数表示，并分类 Neveu-Schwarz 与 Ramond 扇区中的 $\mathfrak{osp}(1|2)$ 极小模型单松弛最高权模。
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…
Ordinary modules for vertex algebras of $\mathfrak{osp}_{1|2n}$
本文证明一般水平下仿射顶点超代数 V^k(osp(1|2n)) 嵌入 sp(2n) 的等变 W-代数与自由费米子的张量积，并由此给出陪集同构的新证明及普通模分解；在可容许水平下进一步证明单代数为子代数的扩张，从而得到普通模范畴是半单刚性顶点张量超范畴且仅有有限多个单对象。
In recent work, we examined the algebraic structure underlying a class of elements supercommuting with realization of the Lie superalgebra $\mathfrak{osp}(1|2)$ inside a generalization of the Weyl Clifford algebra. This generalization contained in particular the deformation by means of Dunkl operators, yielding a rational Cherednik algebra instead of the Weyl algebra. The aim of this work is to show that this is the full supercentralizer, give a (minimal) set of generators, and to describe the relation with the $(\mathrm{Pin}(d),\mathfrak{osp}(2m+1|2n))$ Howe dual pair.
Let $A$ be an associative simple (central) superalgebra over ${\mathbb C}$ and $L$ an invariant linear functional on it (trace). Let $a\mapsto a^t$ be an antiautomorphism of $A$ such that $(a^t)^ t=(-1)^{p(a)}a$, where $p(a)$ is the parity of $a$, and let $L(a^t)=L(a)$. Then $A$ admits a nondegenerate supersymmetric invariant bilinear form $\langle a, b\rangle=L(ab^t)$. For $A=U({\mathfrak{sl}}(2))/{\mathfrak{m}}$, where ${\mathfrak{m}}$ is any maximal ideal of $U({\mathfrak{sl}}(2))$, Leites and I have constructed orthogonal basis in $A$ whose elements turned out to be, essentially, Chebyshev…
Spaces of spinor-valued homogeneous polynomials, and in particular spaces of spinor-valued spherical harmonics, are decomposed in terms of irreducible representations of the symplectic group Sp$( p)$. These Fischer decompositions involve spaces of homogeneous, so-called $\mathfrak{osp}(4|2)$-monogenic polynomials, the Lie superalgebra $\mathfrak{osp}(4|2)$ being the Howe dual partner to the symplectic group Sp$( p)$. In order to obtain Sp$( p)$-irreducibility this new concept of $\mathfrak{osp}(4|2)$-monogenicity has to be introduced as a refinement of quaternionic monogenicity; it is defined …
We prove that the Duflo-Serganova functor $DS_x$ attached to an odd nilpotent element $x$ of $\mathfrak{osp}(m|2n)$ is semisimple, i.e. sends a semisimple representation $M$ of $\mathfrak{osp}(m|2n)$ to a semisimple representation of $\mathfrak{osp}(m-2k|2n-2k)$ where $k$ is the rank of $x$. We prove a closed formula for $DS_x(L(λ))$ in terms of the arc diagram attached to $λ$.
Relaxed highest-weight modules play a central role in the study of many important vertex operator (super)algebras and their associated (logarithmic) conformal field theories, including the admissible-level affine models. Indeed, their structure and their (super)characters together form the crucial input data for the standard module formalism that describes the modular transformations and Grothendieck fusion rules of such theories. In this article, character formulae are proved for relaxed highest-weight modules over the simple admissible-level affine vertex operator superalgebras associated to…
Action of $\mathfrak{osp}(1|2n)$ on polynomials tensor $\mathbb{C}^{0|2n}$
研究 Lie 超代数 $\mathfrak{osp}(1|2n)$ 在多项式与自然模张量上的作用，通过交织算子将张量积表示分解为不可约直和项，并给出基与奇根向量作用公式。
Non-semisimple topological field theory and $\widehat{Z}$-invariants from $\mathfrak{osp}(1 \vert 2)$
本文基于 Lie 超代数 osp(1|2) 构造非半单拓扑场论，并建立了与物理中 Z-不变量的关联。
Maximal subalgebras in the three exceptional simple finite-dimensional Lie superalgebras
本文解决了三个例外简单有限维李超代数的极大分次子代数分类问题，列出了适用分次并给出所有极大分次子代数，且通过判据提升为极大，并补充了分裂分类的第六种情形。
Let $\frak F_ł$ be the space of tensor densities of degree $λ$ on the supercircle $S^{1|1}$. We consider the space $\mathfrak{D}_{λ,μ}^k$ of k-th order linear differential operators from $\frak F_ł$ to $\frak F_μ$ as $\mathfrak{osp}(1|2)$-module and we determine the space $\mathfrak{I}_{λ,μ}^k$ of linear maps on $\mathfrak{D}_{λ,μ}^k$ commuting with the $\mathfrak{osp}(1|2)$-action.
Bethe ansatz equations for orthosymplectic Lie superalgebras and self-dual superspaces
本文研究正交辛李超代数 Bethe ansatz 方程的解，通过再生程序构造解的种群，并证明种群与自对偶超空间中各向同性完全超旗及对称完全分解之间存在典范双射。
On the triviality of inhomogeneous deformations of $\mathfrak{osp}(1|2n)$
研究探讨了$\mathfrak{osp}(1|2n)$的不均匀$γ$-变形的平凡性，证明当$n≥2$时，变形平凡当且仅当所有变形参数消失，通过构造证书明确几何意义。
We establish an explicit formula for the character of an irreducible finite-dimensional representation of $\mathfrak{gl}(m|n)$. The formula is a finite sum with integer coefficients in terms of a basis $\mathcal{E}_μ$ (Euler characters) of the character ring. We prove a simple formula for the behaviour of the ``superversion'' of $\mathcal{E}_μ$ in the $\mathfrak{gl}(m|n)$ and $\mathfrak{osp}(m|2n)$-case under the map $ds$ on the supercharacter ring induced by the Duflo-Serganova cohomology functor $DS$. As an application we get combinatorial formulas for superdimensions, dimensions and $\mathf…
Following Feigin and Fuchs, we compute the first cohomology of the Lie superalgebra $\mathcal{K}(1)$ of contact vector fields on the (1,1)-dimensional real superspace with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. We also compute the same, but $\mathfrak{osp}(1|2)$-relative, cohomology. We explicitly give 1-cocycles spanning these cohomology. We classify generic formal $\mathfrak{osp}(1|2)$-trivial deformations of the $\mathcal{K}(1)$-module structure on the superspaces of symbols of differential operators. We prove that an…
Let $\mathbb{S}$ denote the oscillatory module over the complex symplectic Lie algebra $\mathfrak{g}= \mathfrak{sp}(\mathbb{V}^{\mathbb{C}},ω).$ Consider the $\mathfrak{g}$-module $\mathbb{W}=\bigwedge^{\bullet}(\mathbb{V}^*)^{\mathbb{C}}\otimes \mathbb{S}$ of exterior forms with values in the oscillatory module. We prove that the associative algebra $\hbox{End}_{\mathfrak{g}}(\mathbb{W})$ is generated by the image of a certain representation of the ortho-symplectic Lie super algebra $\mathfrak{osp}(1|2)$ and two distinguished projection operators. The space $\bigwedge^{\bullet}(\mathbb{V}^*)^…
In the first part of this paper, we discuss the classical W-algebra $\mathcal{W}(\mathfrak{g}, F)$ associated with a Lie superalgebra $\mathfrak{g}$ and the nilpotent element $F$ in an $\mathfrak{sl}_2$-triple. We find a generating set of $\mathcal{W}(\mathfrak{g}, F)$ and compute the Poisson brackets between them. In the second part, which is the main part of the paper, we discuss supersymmetric classical W-algebras. We introduce two different constructions of a supersymmetric classical W-algebra $\mathcal{W}(\mathfrak{g}, f)$ associated with a Lie superalgebra $\mathfrak{g}$ and an odd nilpo…
Relative Langlands duality for $\mathfrak{osp}(2n + 1|2n)$
本文研究了 osp(2n + 1|2n) 的相对朗兰兹对偶，证明了 S-对偶关系并提出了相关全球猜想。
We investigate the underlying quantum group symmetry of 2d Liouville and dilaton gravity models, both consolidating known results and extending them to the cases with $\mathcal{N} = 1$ supersymmetry. We first calculate the mixed parabolic representation matrix element (or Whittaker function) of $\text{U}_q(\mathfrak{sl}(2, \mathbb{R}))$ and review its applications to Liouville gravity. We then derive the corresponding matrix element for $\text{U}_q(\mathfrak{osp}(1|2, \mathbb{R}))$ and apply it to explain structural features of $\mathcal{N} = 1$ Liouville supergravity. We show that this matrix…
Orthosymplectic Feigin-Semikhatov duality
本文研究了正交辛类型的Feigin-Semikhatov对称性，通过相对半无限上同调和能级流动扭，建立了W超代数权重模的块等价性，并推导了特征公式。
In this paper we study extensions between finite-dimensional simple modules over classical Lie superalgebras $\mathfrak{gl}(m|n), \mathfrak{osp}(M|2n)$ and $\mathfrak{q}_m$. We consider a simplified version of the extension graph which is produced from the $Ext^1$-graph by identifying representations obtained by parity change and removal of the loops. We give a necessary condition for a pair of vertices to be connected and show that this condition is sufficient in most of the cases. This condition implies that the image of a finite-dimensional simple module under the Duflo-Serganova functor ha…
$F-$Lie algebras are natural generalisations of Lie algebras (F=1) and Lie superalgebras (F=2). When $F>2$ not many finite-dimensional examples are known. In this paper we construct finite-dimensional $F-$Lie algebras $F>2$ by an inductive process starting from Lie algebras and Lie superalgebras. Matrix realisations of $F-$Lie algebras constructed in this way from $\mathfrak{su}(n), \mathfrak{sp}(2n)$ $\mathfrak{so}(n)$ and $\mathfrak{sl}(n|m)$, $\mathfrak{osp}(2|m)$ are given. We obtain non-trivial extensions of the Poincaré algebra by Inönü-Wigner contraction of certain $F-$Lie algebras with…
We construct universal monoidal categories of topological tensor supermodules over the Lie superalgebras $\mathfrak{gl}(V\oplus ΠV)$ and $\mathfrak{osp}(V\oplus ΠV)$ associated with a Tate space $V$. Here $V\oplus ΠV$ is a $\mathbb{Z}/2\mathbb{Z}$-graded topological vector space whose even and odd parts are isomorphic to $V$. We discuss the purely even case first, by introducing monoidal categories$\widehat{\mathbf{T}}_{\mathfrak{gl}(V)}$, $\widehat{\mathbf{T}}_{\mathfrak{o}(V)}$ and $\widehat{\mathbf{T}}_{\mathfrak{sp}(V)}$, and show that these categories are anti-equivalent to respective pre…
We compute the first cohomology of the ortosymplectic Lie superalgebra $\mathfrak{osp}(1|2)$ on the (1,1)-dimensional real superspace with coefficients in the superspace $\frak{D}_{λ,ν;μ}$ of bilinear differential operators acting on weighted densities. This work is the simplest superization of a result by Bouarroudj [Cohomology of the vector fields Lie algebras on $\mathbb{R}\mathbb{P}^1$ acting on bilinear differential operators, International Journal of Geometric Methods in Modern Physics (2005), {\bf 2}; N 1, 23-40].
Trialities of $\mathcal{W}$-algebras are isomorphisms between the affine cosets of three different $\mathcal{W}$-(super)algebras, and were first conjectured in the physics literature by Gaiotto and Rapčák. In this paper we prove trialities among eight families of $\mathcal{W}$-(super)algebras of types $B$, $C$, and $D$. The key idea is to identify the affine cosets of these algebras with one-parameter quotients of the universal two-parameter even spin $\mathcal{W}_{\infty}$-algebra which was recently constructed by Kanade and the second author. Our result is a vast generalization of both Feigi…
Dynkin's classification of maximal subalgebras of simple finite dimensional complex Lie algebras is generalized to linear Lie superalgebras. Namely, the maximal non-simple irreducible subalgebras of $\mathfrak{gl}(p|q), \mathfrak{q}(n), \mathfrak{sl}(p|q), \mathfrak{osp}(m|2n), \mathfrak{pe}(n)$, and $\mathfrak{spe}(n)$ are classified.
Higher-spin symmetry in the $\mathfrak{sl}_3$ boundary Toda conformal field theory II: Singular vectors and BPZ equations
本文基于概率定义和前期框架，计算 $\mathfrak{sl}_3$ 边界 Toda 共形场论中的奇异向量，导出高阶运动方程及 BPZ 型微分方程，为严格推导结构常数奠定基础。
We construct a minimal representation of the orthosymplectic Lie supergroup $OSp(p,q|2n)$, generalising the Schrödinger model of the minimal representation of $O(p,q)$ to the super case. The underlying Lie algebra representation is realized on functions on the minimal orbit inside the Jordan superalgebra associated with $\mathfrak{osp}(p,q|2n)$, so that our construction is in line with the orbit philosophy. Its annihilator is given by a Joseph-like ideal for $\mathfrak{osp}(p,q|2n)$, and therefore the representation is a natural generalization of a minimal representations to the context of Lie…
In this short note we show that the Bruhat cells in real normal forms of semisimple Lie algebras enjoy the same property as their complex analogs: for any two elements $w$, $w'$ in the Weyl group $W(\mathfrak g)$, the corresponding real Bruhat cell $X_w$ intersects with the dual Bruhat cell $Y_{w'}$ iff $w\prec w'$ in the Bruhat order on $W(\mathfrak g)$. Here $\mathfrak g$ is a normal real form of a semisimple complex Lie algebra $\mathfrak g_\mathbb C$. Our reasoning is based on the properties of the Toda flows rather than on the analysis of the Weyl group action and geometric considerations…
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