Nicoletta Cantarini, Fabrizio Caselli, Victor Kac
We construct a duality functor on the category of continuous representations of linearly compact Lie superalgebras, using representation theory of Lie conformal superalgebras. We compute the dual representations of the generalized Verma modules.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Let A_k denote the twisted group algebra of the symmetric group S_k, whose representations correspond to the nonlinear projective representations of S_k. We establish a duality relation between A_k and a Lie superalgebra q(n), sometimes called the ``queer'' Lie superalgebra. This duality clarifies Schur's formula, in the sense that the Schur-Weyl duality clarifies Frobenius' formula.
We establish a new Howe duality between a pair of quantum queer superalgebras $(\mathrm{U}_{q^{-1}}(\mathfrak{q}_n), \mathrm{U}_q(\mathfrak{q}_m))$. The key ingredient is the construction of a non-commutative analogue $\mathcal{A}_q(\mathfrak{q}_n,\mathfrak{q}_m)$ of the symmetric superalgebra $S(\mathbb{C}^{mn|mn})$ with the use of quantum coordinate queer superalgebra. It turns out that this superalgebra is equipped with a $\mathrm{U}_{q^{-1}}(\mathfrak{q}_n)\otimes\mathrm{U}_q(\mathfrak{q}_m)$-supermodule structure that admits a multiplicity-free decomposition. We also show that the $(\math…
We develop a new method to solve the irreducible character problem for a wide class of modules over the general linear superalgebra, including all the finite-dimensional modules, by directly relating the problem to the classical Kazhdan-Lusztig theory. We further verify a parabolic version of a conjecture of Brundan on the irreducible characters in the BGG category $\mc{O}$ of the general linear superalgebra. We also prove the super duality conjecture.
Howe's duality is considered from a unifying point of view based on Lie superalgebras. New examples are offered. In particular, we construct several simplest spinor-oscillator representations and compute their highest weights for the 'stringy' Lie superalgebras (i.e., Lie superalgebras of complex vector fields (or their nontrivial central extensions) on the supercircle $S^{1|n}$ and its two-sheeted cover associated with the Möbius bundle).
We construct a duality functor in the category of continuous representations of the Lie superalgebra E(4,4), the only exceptional simple linearly compact Lie superalgebra, for which it wasn't known. This is achieved by constructing a Lie conformal superalgebra of type (4,4), for which E(4,4) is the annihilation algebra. Along the way we obtain an explicit realization of E(4,4) by vector fields on a (4|4)-dimensional supermanifold.
In the present paper we classify all irreducible continuous representations of the simple linearly compact n-Lie superalgebra of type W. The classification is based on a bijective correspondence between the continuous representations of the n-Lie algebras W^n and continuous representations of the Lie algebra of Cartan type W_{n-1}, on which some two-sided ideal acts trivially.
Using Howe duality we compute explicitly Kostant-type homology groups for a wide class of representations of the infinite-dimensional Lie superalgebra $\hat{\frak{gl}}_{\infty|\infty}$ and its classical subalgebras at positive integral levels. We also obtain Kostant-type homology formulas for the Lie algebra $ widehat{\frak{gl}}_\infty$ at negative integral levels. We further construct resolutions in terms of generalized Verma modules for these representations.
In the present paper we classify all irreducible continuous representations of the simple linearly compact n-Lie superalgebra of type S. The classification is based on a bijective correspondence between the continuous representations of the n-Lie algebras S^n and continuous representations of the Lie algebra of Cartan type S, on which some two-sided ideal acts trivially.
We apply the super duality formalism recently developed by the authors to obtain new equivalences of various module categories of general linear Lie superalgebras. We establish the correspondence of standard, tilting, and simple modules, as well as the identification of the u-homology groups, under these category equivalences. As an application, we obtain a complete solution of the irreducible character problem for some new parabolic BGG categories of gl(m|n)-modules, including the full BGG category of gl(m|2)-modules, in terms of type A Kazhdan-Lusztig polynomials.
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.
Super Vust theorem and Schur-Sergeev duality for principal finite $W$-superalgebras
该研究针对一般线性李超代数提出超Vust定理，并以此为工具得到主有限$W$-超代数的Schur-Sergeev对偶，部分推广了Brundan-Kleshchev高阶Schur-Weyl对偶。
We give a proof of a Schur-Weyl duality statement between the Brauer algebra and the ortho-symplectic Lie superalgebra $\mathfrak{osp}(V)$.
Two super-analogs of the Schur-Weyl duality are considered: the duality of actions in $(\mathbb{C}^{m|n})^{\otimes N}$ of the Lie superalgebra $\mathfrak{gl}(m,n)$ and the symmetric group $S_N$, and the duality of actions of the Lie superalgebra $Q(n)$ and a certain finite group $Se(N)$ in $(\mathbb{C}^{n|n})^{\otimes N}$. We construct an isomorphism of symmetric and universal enveloping algebras of these Lie superalgebras called special symmetrization. Using this isomorphism of vector spaces we describe explicitly the duality between the centers of the corresponding universal enveloping algeb…
We study a dual pair of general linear Lie superalgebras in the sense of R. Howe. We give an explicit multiplicity-free decomposition of a symmetric and skew-symmetric algebra (in the super sense) under the action of the dual pair and present explicit formulas for the highest weight vectors in each isotypic subspace of the symmetric algebra. We give an explicit multiplicity-free decomposition into irreducible $gl(m|n)$-modules of the symmetric and skew-symmetric algebras of the symmetric square of the natural representation of $gl(m|n)$. In the former case we find as well explicit formulas for…
We formulate and establish a super duality which connects parabolic categories $O$ between the ortho-symplectic Lie superalgebras and classical Lie algebras of $BCD$ types. This provides a complete and conceptual solution of the irreducible character problem for the ortho-symplectic Lie superalgebras in a parabolic category $O$, which includes all finite-dimensional irreducible modules, in terms of classical Kazhdan-Lusztig polynomials.
We establish classical and categorical Howe dualities between the Lie superalgebras $\mathfrak{p}(m)$ and $\mathfrak{p}(n)$, for $m,n \geq 1$. We also describe a presentation via generators and relations as well as a Kostant $\mathbb{Z}$-form for the universal enveloping superalgebra $U(\mathfrak{p}(m))$.
Generalizing the super duality formalism for finite-dimensional Lie superalgebras of type $ABCD$, we establish an equivalence between parabolic BGG categories of a Kac-Moody Lie superalgebra and a Kac-Moody Lie algebra. The characters for a large family of irreducible highest weight modules over a symmetrizable Kac-Moody Lie superalgebra are then given in terms of Kazhdan-Lusztig polynomials for the first time. We formulate a notion of integrable modules over a symmetrizable Kac-Moody Lie superalgebra via super duality, and show that these integrable modules form a semisimple tensor subcategor…
$q$-deformed Howe duality for orthosymplectic Lie superalgebras
本文给出正交辛李超代数与正交群或辛群配对Howe对偶的$q$-模拟，构造量子包络代数与$\imath$量子群的交换作用，并恢复经典对偶。
We provide a new proof of the super duality equivalence between infinite-rank parabolic BGG categories of general linear Lie (super) algebras conjectured by Cheng and Wang and first proved by Cheng and Lam. We do this by establishing a new uniqueness theorem for tensor product categorifications motivated by work of Brundan, Losev, and Webster. Moreover we show that these BGG categories have Koszul graded lifts and super duality can be lifted to a graded equivalence.
We introduce a new family of superalgebras $\overrightarrow{B}_{r,s}$ for $r, s \ge 0$ such that $r+s>0$, which we call the walled Brauer superalgebras, and prove the mixed Scur-Weyl-Sergeev duality for queer Lie superalgebras. More precisely, let $\mathfrak{q}(n)$ be the queer Lie superalgebra, ${\mathbf V} =\mathbb{C}^{n|n}$ the natural representation of $\mathfrak{q}(n)$ and ${\mathbf W}$ the dual of ${\mathbf V}$. We prove that, if $n \ge r+s$, the superalgebra $\overrightarrow{B}_{r,s}$ is isomorphic to the supercentralizer algebra $_{\mathfrak{q}(n)}({\mathbf V}^{\otimes r} \otimes {\mat…
Schur-Weyl duality for quantum toroidal superalgebras
本文建立了双重仿射 Hecke 代数与量子环面超代数之间的 Schur-Weyl 型对偶，将 Vasserot-Varagnolo [VV96] 的经典结果推广到了超情形。
We establish a duality relation between one of the twisted group algebras of the hyperoctahedral groupf H_k and a Lie superalgebra q(n_0) \oplus q(n_1) for any integers k and n_0, n_1, where q(n_0) and q(n_1) denote the ``queer'' Liesuperalgebras. Note that this twisted group algebra \B'_k belongs to a different cocycle from the one \B_k used by A. N. Sergeev in [8] and by the present author in [11]. We will use the supertensor product \C_k \otimes \B'_k of the 2^k-dimensional Clifford algebra \C_k and \B'_k, as an intermediary for establishing our duality. We show that the algebra \C_k \otime…
The Schur-Weyl duality and Invariants for classical Lie superalgebras
本文基于Schur-Weyl对偶性的超类比，统一刻画经典李超代数的不变量，建立其张量代数等的$\frak{g}$-不变量及关系，证明投影限制满射性以确定中心生成元，并给出$\frak{Z}(\frak{p}_n)$平凡性的新证明。
Super duality for Whittaker modules and finite $W$-algebras
本文通过建立Whittaker模块与有限$W$-代数之间的超双曲性，揭示了无限秩极限下模块范畴的等价关系。
Degenerating the quantum queer Schur superalgebra ${\mathcal{Q}_q(n,r; R)}$ to the case $q=1$, the queer Schur superalgebra ${\mathcal{Q}(n,r)}$ is obtained. In this article, we reconstruct the universal enveloping algebra ${U({\mathfrak{q}_n})}$ of the queer Lie superalgebra ${\mathfrak{q}_n}$ via ${\mathcal{Q}(n,r)}$, and achieve another explanation of the Schur-Weyl-Sergeev duality. Finally, we depict the Schur-Weyl-Sergeev duality over $\mathbb{Z}$.
In this work infinitesimal deformations of the model filiform $\mathbb{Z}_2 \times \mathbb{Z}_2$-color Lie superalgebra have been studied. All the filiform $\mathbb{Z}_2 \times \mathbb{Z}_2$-color Lie superalgebras can be obtained by means of infinitesimal deformations, hence the importance of these. Thus, in particular, we give a family of filiform $\mathbb{Z}_2 \times \mathbb{Z}_2$-color Lie superalgebras via linearly integrable deformations.
We study the Howe dualities involving the reductive dual pairs $(O(d),spo(2m|2n))$ and $(Sp(d),osp(2m|2n))$ on the (super)symmetric tensor of $\C^d\otimes\C^{m|n}$. We obtain complete decompositions of this space with respect to their respective joint actions. We also use these dualities to derive a character formula for these irreducible representations of $spo(2m|2n)$ and $osp(2m|2n)$ that appear in these decompositions.
We introduce a semisimple tensor category $\mc{O}^{int}_q(m|n)$ of modules over an quantum ortho-symplectic superalgebra. It is a natural counterpart of the category of finitely dominated integrable modules over the quantum classical (super) algebra of type $B_{m+n}$, $C_{m+n}$, $D_{m+n}$ or $B(0,m+n)$ from a viewpoint of super duality. We classify the irreducible modules in $\mc{O}^{int}_q(m|n)$ and show that an irreducible module in $\mc{O}^{int}_q(m|n)$ has a unique crystal base in case of type $B$ and $C$. An explicit description of the crystal graph is given in terms of a new combinatoria…
We study tilting and projective-injective modules in a parabolic BGG category $\mathcal O$ for an arbitrary classical Lie superalgebra. We establish a version of Ringel duality for this type of Lie superalgebras which allows to express the characters of tilting modules in terms of those of simple modules in that category. We also obtain a classification of projective-injective modules in the full BGG category $\mathcal O$ for all simple classical Lie superalgebras. We then classify and give an explicit combinatorial description of parabolic subalgebras of the periplectic Lie superalgebras and …
We explain how Lie superalgebras of types gl and osp provide a natural framework generalizing the classical Schur and Howe dualities. This exposition includes a discussion of super duality, which connects the parabolic categories O between classical Lie superalgebras and Lie algebras. Super duality provides a conceptual solution to the irreducible character problem for these Lie superalgebras in terms of the classical Kazhdan-Lusztig polynomials.
In this paper we introduce generalized Spencer cohomology for finite depth Z-graded Lie (super)algebras. We develop a method of finding filtered deformations of such Z-graded Lie (super)algebras based on this cohomology. As an application we determine all simple filtered deformations of certain Z-graded Lie superalgebras classified in [K3], thus completing the last step in the classification of simple infinite-dimensional linearly compact Lie superalgebras.
Duality properties for induced and coinduced representations in positive characteristic
本文在正特征情况下研究了李超代数诱导与余诱导表示的对偶性质，并探讨了受限李超代数中诱导与余诱导表示之间的联系。
We use the language of Lie pseudoalgebras to gain information about the representation theory of the simple infinite-dimensional linearly compact Lie superalgebra of exceptional type $E(5,10)$. This technology allows us to prove that the degree of singular vectors in minimal Verma modules is $\leq 14$. A few technical adjustments allow us to refine the bound, proving that the degree must always be $\leq 12$ and it is actually, except for a finite number of cases, $\leq 10$.
The characteristic identity formalism discussed in our recent articles is further utilized to derive matrix elements of type 2 unitary irreducible $gl(m|n)$ modules. In particular, we give matrix element formulae for all gl(m|n) generators, including the non-elementary generators, together with their phases on finite dimensional type 2 unitary irreducible representations. Remarkably, we find that the type 2 unitary matrix element equations coincide with the type 1 unitary matrix element equations for non-vanishing matrix elements up to a phase.
There is a surprising isomorphism between the quantised universal enveloping algebras of osp(1|2n) and so(2n+1). This same isomorphism emerged in recent work of Mikhaylov and Witten in the context of string theory as a T-duality composed with an S-duality. We construct similar Hopf superalgebra isomorphisms for families of pairs of quantum affine superalgebras. An immediate consequence is that the representation categories of the quantum affine superalgebras in each pair are equivalent as strict tensor categories.
Classification of degenerate Verma modules over $E(4,4)$
本文对$E(4,4)$上退化Verma模块进行分类，完善了例外线性紧李超代数上Verma模块的描述，退化模与它们之间的映射产生无限双边复形，可视为de Rham复形的推广。
Let g=g_{0} \oplus g_{1} be a classical Lie superalgebra and F be the category of finite dimensional g-supermodules which are semisimple over g_{0}. In this paper we investigate the homological properties of the category F. In particular we prove that F is self-injective in the sense that all projective supermodules are injective. We also show that all supermodules in F admit a projective resolution with polynomial rate of growth and, hence, one can study complexity in F. If g is a Type I Lie superalgebra we introduce support varieties which detect projectivity and are related to the associate…
We study the ideal of maximal minors in Littlewood varieties, a class of quadratic complete intersections in spaces of matrices. We give a geometric construction for a large class of modules, including all powers of this ideal, and show that they have a linear free resolution over the complete intersection and that their Koszul dual is an infinite-dimensional irreducible representation of the orthosymplectic Lie superalgebra. We calculate the algebra of cohomology operators acting on this free resolution. We prove analogous results for powers of the ideals of maximal minors in the variety of l…
We classify all continuous degenerate irreducible modules over the exceptional linearly compact Lie superalgebra E(1, 6), and all finite degenerate irreducible modules over the exceptional Lie conformal superalgebra CK6, for which E(1, 6) is the annihilation algebra.
Making use of a Howe duality involving the infinite-dimensional Lie superalgebra $\hgltwo$ and the finite-dimensional group $GL_l$ we derive a character formula for a certain class of irreducible quasi-finite representations of $\hgltwo$ in terms of hook Schur functions. We use the reduction procedure of $\hgltwo$ to $\hat{gl}_{n|n}$ to derive a character formula for a certain class of level 1 highest weight irreducible representations of $\hat{gl}_{n|n}$, the affine Lie superalgebra associated to the finite-dimensional Lie superalgebra $gl_{n|n}$. These modules turn out to form the complete s…
Affine super Yangians and deformed double current superalgebras
该研究将Schur-Weyl对偶相关结果推广到特殊线性李超代数$\mathfrak{sl}_{m|n}$的超Yangian及其仿射版本，还首次提出该代数的形变双流超代数并完成同类推广。
Quiver Hecke--Clifford superalgebras and R-matrices
本文建立了 quiver Hecke–Clifford 超代数的表示理论基础，并基于 Kwon–Lee 的构造，建立了 queer 李超代数的量子仿射类比与 A 型 quiver Hecke–Clifford 超代数之间的 Schur–Weyl 对偶。
We prove Feigin-Frenkel type dualities between subregular W-algebras of type A, B and principal W-superalgebras of type $\mathfrak{sl}(1|n), \mathfrak{osp}(2|2n)$. The type A case proves a conjecture of Feigin and Semikhatov. Let $(\mathfrak{g}_1,\mathfrak{g}_2) = (\mathfrak{sl}_{n+1},\mathfrak{sl}(1|n+1))$ or $(\mathfrak{so}_{2n+1}, \mathfrak{osp}(2|2n))$ and let $r$ be the lacity of $\mathfrak{g}_1$. Let k be a complex number and $\ell$ defined by $r(k+h^\vee_1)(\ell+h^\vee_2)=1$ with $h^\vee_i$ the dual Coxeter numbers of the $\mathfrak g_i$. Our first main result is that the Heisenberg cos…
We formulate a general super duality conjecture on connections between parabolic categories O of modules over Lie superalgebras and Lie algebras of type A, based on a Fock space formalism of their Kazhdan-Lusztig theories which was initiated by Brundan. We show that the Brundan-Kazhdan-Lusztig (BKL) polynomials for Lie superalgebra gl(m|n) in our parabolic setup can be identified with the usual parabolic Kazhdan-Lusztig polynomials. We establish some special cases of the BKL conjecture on the parabolic category O of gl(m|n)-modules and additional results which support the BKL conjecture and su…
We establish a direct connection between the representation theories of Lie algebras and Lie superalgebras (of type A) via Fock space reformulations of their Kazhdan-Lusztig theories. As a consequence, the characters of finite-dimensional irreducible modules of the general linear Lie superalgebra are computed by the usual parabolic Kazhdan-Lusztig polynomials of type A. In addition, we establish closed formulas for canonical and dual canonical bases for the tensor product of any two fundamental representations of type A.
The symmetric group S_n possesses a nontrivial central extension, whose irreducible representations, different from the irreducible representations of S_n itself, coincide with the irreducible representations of a certain algebra A_n. Recently M.~Nazarov realized irreducible representations of A_n and Young symmetrizers by means of the Howe duality between the Lie superalgebra q(n) and the Hecke algebra H_n, the semidirect product of S_n with the Clifford algebra C_n on n indeterminates. Here I construct one more analog of Young symmetrizers in H_n as well as the analogs of Specht modules for …
The exceptional infinite-dimensional linearly compact simple Lie superalgebra ${E}(5,10)$, which Kac believes, is the algebra of symmetries of the ${SU}_{5}$ Grand Unified Model. In this paper, we give a proof of Kac and Rudakov's conjecture about the classification of all the degenerate generalized Verma module over ${E}(5,10)$. Also, we work out all the nontrivial singular vectors degree by degree. It is a potential that the representation theory of ${E}(5,10)$ will shed new light on various features of the the ${SU}_{5}$ Grand unified model.
We introduce categories of homogeneous strict polynomial functors, $\Pol^\I_{d,\k}$ and $\Pol^\II_{d,\k}$, defined on vector superspaces over a field $\k$ of characteristic not equal 2. These categories are related to polynomial representations of the supergroups $GL(m|n)$ and Q(n), respectively. In particular, we prove an equivalence between $\Pol^\I_{d,\k}$, $\Pol^\II_{d,\k}$ and the category of finite dimensional supermodules over the Schur superalgebra $\Sc(m|n,d)$, $\Qc(n,d)$ respectively provided $m,n \ge d$. We also discuss some aspects of Sergeev duality from the viewpoint of the categ…
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…
Imaginary Schur-Weyl duality for quiver Hecke superalgebras
本文研究箭图 Hecke 超代数的虚尖代数，通过构造非负分次的 Morita 等价代数 C(n,d)，将其表示论归结为经典 Schur 代数，从而用多重分拆典范分类不可约模。
In the first part of my talk I will explain a solution to the extension of Lie's problem on classification of "local continuous transformation groups of a finite-dimensional manifold" to the case of supermanifolds. (More precisely, the problem is to classify simple linearly compact Lie superalgebras, i.e. toplogical Lie superalgebras whose underlying space is a topological product of finite-dimensional vector spaces). In the second part I will explain how this result is used in a classification of superconformal algebras. The list consists of affine superalgebras and certain super extensions o…
Unitary branching rules for the general linear Lie superalgebra
该研究利用Howe对偶与Kac模分支规则，建立了一般线性李超代数$\frak{gl}_{m|n}$有限维酉单模的最高权形式分支规则，并推导了其对偶的2型酉单模的分支规则。
Feigin-Semikhatov duality at the critical level
研究临界水平下Feigin-Semikhatov对偶性，通过构造函子实现顶点代数范畴的块状等价，揭示次正则W-代数与主W-超代数的结构关系。
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…
In this paper, a notion of cyclotomic (or level $k$) walled Brauer algebras $\mathscr B_{k, r, t}$ is introduced for arbitrary positive integer $k$. It is proven that $\mathscr B_{k, r, t}$ is free over a commutative ring with rank $k^{r+t}(r+t)!$ if and only if it is admissible. Using super Schur-Weyl duality between general linear Lie superalgebras $\mathfrak{gl}_{m|n}$ and $\mathscr B_{2, r, t}$, we give a classification of highest weight vectors of $\mathfrak{gl}_{m|n}$-modules $M_{pq}^{rt}$, the tensor products of Kac-modules with mixed tensor products of the natural module and its dual. …
Verma Bases for finite dimensional Representations of the orthosymplectic Lie superalgebra $\mathfrak{spo}(4|1)$
本文定义了正交辛李超代数$\mathfrak{spo}(4|1)$有限维不可约表示的Verma向量系统，通过Kashiwara-Nakashima表的条件，证明其线性无关性，并展示其作为该表示的基。
We establish a new Howe duality between a pair of two queer Lie superalgebras (q(m),q(n)). This gives a representation theoretic interpretation of a well-known combinatorial identity for Schur Q-functions. We further establish the equivalence between this new Howe duality and the Schur-Sergeev duality between q(n) and a central extension $\Hy_k$ of the hyperoctahedral group H_k. We show that the zero-weight space of a q(n)-module with highest weight $λ$ given by a strict partition of n is an irreducible module over the finite group $\Hy_n$ parameterized by $λ$. We also discuss some consequence…
Let $U_q(\mathfrak{g})$ be the quantized superalgebra of $\mathfrak{g}=\mathfrak{gl}(k_1|\ell_1)\oplus\cdots\oplus\mathfrak{gl}(k_m|\ell_m)$ and $H_{m,n}(q,\mathbf{Q})$ the cyclotomic Hecke algebra of type $G(m,1,n)$. We define a right $H_{m,n}(q,\mathbf{Q})$-action on the $n$-fold tensor (super)space of the vector representation of $U_q(\mathfrak{g})$ and prove the Schur--Weyl reciprocity between $U_q(\mathfak{g})$ and $H_{m,n}(q,\mathbf{Q})$.
We study the cohomology of Lie superalgebras for the full complex of forms: superforms, pseudoforms and integral forms. We use the technique of spectral sequences to abstractly compute the Chevalley-Eilenberg cohomology. We first focus on the superalgebra $\mathfrak{osp}(2|2)$ and show that there exist non-empty cohomology spaces among pseudoforms related to sub-superalgebras. We then extend some classical theorems by Koszul, as to include pseudoforms and integral forms. Further, we conjecture that the algebraic Poincaré duality extends to Lie superalgebras, as long as all the complexes of for…
Khovanov algebras for the periplectic Lie superalgebras
该研究针对奇正交李超代数$\mathfrak{p}(n)$这一了解较少的经典单李超代数，通过Schur–Weyl对偶，借助超Brauer代数的幂等版本构造类Khovanov的投射生成元自同态环，给出平移函子的图表描述并研究其作用，为后续表示分类、上同调计算及分次性质证明提供基础。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。