We construct all the irreducible representations of spin quiver Hecke algebras for orthosymplectic Lie superalgebras $osp(1|2n),$ and show that their highest weights are given by the dominant words. We use the dominant Lyndon words to construct the cuspidal modules and show that the irreducible representations are the simple heads of standard representations constructed by induction from the cuspidal modules.
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.
Drinfeld Realization for Quantum Affine Orthosymplectic Superalgebras
Instituto de Ciências Exatas, Universidade Federal de Minas Gerais（米纳斯吉拉斯联邦大学精确科学学院）；Shenzhen International Center for Mathematics, Southern University of Science and Technology（南方科技大学深圳国际数学中心）
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 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.
Orthosymplectic Lie superalgebras are fundamental symmetries in modern physics, such as massive supergravity. However, their representations are far from being thoroughly understood. In the present paper, we completely determine the structure of their various supersymmetric polynomial representations obtained by swapping bosonic multiplication operators and differential operators in the canonical supersymmetric polynomial representations. In particular, we obtain certain new infinite-dimensional irreducible representations and new composition series of indecomposable representations for these …
We prove an Amitsur-Levitzki type theorem for the Lie superalgebras osp(1,2n) inspired by Kostant's cohomological interpretation of the classical theorem. We show that the Lie superalgebras gl(p,q) cannot satisfy an Amitsur-Levitzki type super identity if p, q are non zero and conjecture that neither can any other classical simple Lie superalgebra with the exception of osp(1,2n).
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…
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 $λ$.
The composition factors and their multiplicities are determined for generalised Verma modules over the orthosymplectic Lie superalgebra osp(k|2). The results enable us to obtain explicit formulae for the formal characters and dimensions of the finite-dimensional irreducible modules. Applying these results, we also compute the first and second cohomology groups of the Lie superalgebra with coefficients in finite-dimensional Kac modules and irreducible modules.
We introduce the orthosymplectic superalgebra osp(m|2n) as the algebra of Killing vector fields on Riemannian superspace R^{m|2n} which stabilize the origin. The Laplace operator and norm squared on R^{m|2n}, which generate sl(2), are orthosymplectically invariant, therefore we obtain the Howe dual pair (osp(m|2n),sl(2)). We study the osp(m|2n)-representation structure of the kernel of the Laplace operator. This also yields the decomposition of the supersymmetric tensor powers of the fundamental osp(m|2n)-representation under the action of sl(2) x osp(m|2n). As a side result we obtain informat…
We utilise characteristic identities to construct eigenvalue formulae for invariants and reduced matrix elements corresponding to irreducible representations of osp(m|n). In presenting these results, we further develop our programme of constructive representation theory via characteristic identities.
We introduce a new family of superalgebras which should be considered as a super version of the Khovanov-Lauda-Rouquier algebras. Let $I$ be the set of vertices of a Dynkin diagram with parity. To this data, we associate a family of graded superalgebras, the quiver Hecke superalgebras. When there are no odd vertices, these algebras are nothing but the usual Khovanov-Lauda-Rouquier algebras. We then define another family of graded superalgebras, the quiver Hecke-Clifford superalgebras, and show that they are weakly Morita superequivalent to the quiver Hecke superalgebras. Moreover, we prove tha…
We introduce and obtain multimode paraboson coherent states. In appropriate subspaces these coherent states provide a decomposition of unity where the measure, when expressed using the cat-type states, is positive definite. Bicoherent states where the mutually commuting lowering operators are diagonalized are also obtained. Matrix elements in the coherent state basis are calculated.
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…
A Schroedinger type equation on the superspace R^{D|2n} is studied, which involves a potential inversely proportional to the negative of the osp(D|2n) invariant "distance" away from the origin. An osp(2,D+1|2n) dynamical supersymmetry for the system is explicitly constructed, and the bound states of the system are shown to form an irreducible highest weight module for this superalgebra. A thorough understanding of the structure of the irreducible module is obtained. This in particular enables the determination of the energy eigenvalues and the corresponding eigenspaces as well as their respect…
In this paper we give Iwahori-Hecke type algebras H_q(g) associated with the Lie superalgebras g=A(m,n), B(m,n), C(n) and D(m,n). We classify the irreducible representations of H_q(g) for generic q.
In this paper, we introduce the sign q-permutation representation of the Iwahori-Hecke algebra on the tensor space of the graded vector space. We establish Schur-Weyl reciprocity between the quantum general super Lie algebra and the Iwahori-Hecke algebra. Our result includes Schur-Weyl reciprocity of both the super case and the quantum case, and unifies those two important cases.
We construct two infinite-dimensional irreducible representations for $D(2,1;α)$: a Schrödinger model and a Fock model. Further, we also introduce an intertwining isomorphism. These representations are similar to the minimal representations constructed for the orthosymplectic Lie supergroup and for Hermitian Lie groups of tube type. The intertwining isomorphism is the analogue of the Segal-Bargmann transform for the orthosymplectic Lie supergroup and for Hermitian Lie groups of tube type.
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…
Brundan and Kleshchev showed that some parts of the representation theory of the affine Hecke-Clifford superalgebras and its finite-dimensional "cyclotomic" quotients are controlled by the Lie theory of type $A^{(2)}_{2l}$ when the quantum parameter $q$ is a primitive $(2l+1)$-th root of unity. We show in this paper that similar theorems hold when $q$ is a primitive $4l$-th root of unity by replacing the Lie theory of type $A^{(2)}_{2l}$ with that of type $D^{(2)}_{l}$.
We introduce the spinor representations for osp(m|2n). These generalize the spinors for so(m) and the symplectic spinors for sp(2n) and correspond to representations of the supergroup with supergroup pair (Spin(m) x Mp(2n),osp(m|2n)). We prove that these spinor spaces are uniquely characterized as the completely pointed osp(m|2n)-modules. Then the tensor product of this representation with irreducible finite dimensional osp(m|2n)-modules is studied. Therefore we derive a criterion for complete reducibility of tensor product representations. We calculate the decomposition into irreducible osp(m…
It is shown that the Clifford superalgebra Cl(n|m) generated by m pairs of Bose operators (odd elements) anticommuting with n pairs of Fermi operators (even elements) can be deformed to Cl_q(n|m) such that the latter is a homomorphic image of the quantum superalgebra U_q[osp(2n+1|2m)]. The Fock space F(n|m) of Cl_q(n|m) is constructed. At q being a root of unity (q=\exp ({iπl/k})) q-bosons (and q-fermions) are operators acting in a finite-dimensional subspace F_{l/k}(n|m) of F(n|m). Each F_{l/k}(n|m) is turned through the above mentioned homomorphism into an irreducible (root of unity) U_q[osp…
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…
We decompose the fibers of the Springer resolution for the odd nilcone of the Lie superalgebra $\osp(2n+1,2n)$ into locally closed subsets. We use this decomposition to prove that almost all fibers are connected. However, in contrast with the classical Springer fibers, we prove that the fibers can be disconnected and non equidimensional.
We propose to use interpolation categories to study PBW deformations, and demonstrate this idea for the orthosymplectic supergroups. Employing a combinatorial calculus based on pseudographs and partitions which we derive from a suitable Jacobi identity, we classify PBW deformations in (quotients of) Deligne's interpolation categories for the orthosymplectic groups. As special cases, our classification recovers families of infinitesimal Hecke algebras found by Etingof-Gan-Ginzburg (2005) for the symplectic groups and by Tsymbaliuk (2015) for the orthogonal groups together with their respective …
We give a complete description of the finite-dimensional irreducible representations of the Yangian associated with the orthosymplectic Lie superalgebra $\frak{osp}_{1|2}$. The representations are parameterized by monic polynomials in one variable, they are classified in terms of highest weights. We give explicit constructions of a family of elementary modules of the Yangian and show that a wide class of irreducible representations can be produced by taking tensor products of the elementary modules.
We give an elementary proof of the first fundamental theorem of the invariant theory for the orthosymplectic supergroup by generalising the method of Atiyah, Bott and Patodi to the supergroup context. We use methods from super-algebraic geometry to convert invariants of the orthosymplectic supergroup into invariants of the corresponding general linear supergroup on a different space. In this way, a super Schur-Weyl-Brauer duality is established between the orthosymplectic supergroup of superdimension $(m|2n)$ and the Brauer algebra with parameter $m-2n$. The result may be interpreted in terms …
Ian Grojnowski has developed a purely algebraic way to connect the representation theory of affine Hecke algebras at an (l+1)-th root of unity to the highest weight theory of the affine Kac-Moody algebra of type A_l^(1). The present article is devoted to extending Grojnowski's machinery to the twisted case: we replace the affine Hecke algebras with the affine Hecke-Clifford superalgebras of Jones and Nazarov, and the Kac-Moody algebra A_l^(1) with the twisted algebra A_2l^(2). In particular, we obtain an algebraic construction purely in terms of the representation theory of Hecke-Clifford supe…
We define and study RoCK blocks for double covers of symmetric groups. We prove that RoCK blocks of double covers are Morita equivalent to standard `local' blocks. The analogous result for blocks of symmetric groups, a theorem of Chuang and Kessar, was an important step in Chuang and Rouquier ultimately proving Broué's abelian defect group conjecture for symmetric groups. Indeed we prove Broué's conjecture for the RoCK blocks defined in this article. Our methods involve translation into the quiver Hecke superalgebras setting using the Kang-Kashiwara-Tsuchioka isomorphism and categorification m…
We construct the finite dimensional simple integral modules for the (degenerate) affine Hecke-Clifford algebra (AHCA). Our construction includes an analogue of Zelevinsky's segment representations, a complete combinatorial description of the simple calibrated modules, and a classification of the simple integral modules. Additionally, we construct an analogue of the Arakawa-Suzuki functor for the Lie superalgebra of type Q.
As a natural generalization quantum Schur algebras associated with the Hecke algebra of the symmetric group, we introduce the quantum Schur superalgebra of type Q associated with the Hecke-Clifford superalgebra, which, by definition, is the endomorphism algebra of the induced module over the Hecke-Clifford superalgebra from certain permutation modules over the Hecke algebra of the symmetric group. We will describe certain integral bases for these superalgebras in terms of matrices and will establish the base change property for them. We will also identify the quantum Schur superalgebra of type…
In this paper, we investigate the supercategories consisting of supermodules over quiver Hecke superalgebras and cyclotomic quiver Hecke superalgebras. We prove that these supercategories provide a supercategorification of a certain family of quantum superalgebras and their integrable highest weight modules. We show that, by taking a specialization, we obtain a supercategorification of quantum Kac-Moody superalgebras and their integrable highest weight modules.
On the generalized graded cellular bases for cyclotomic quiver Hecke-Clifford superalgebras
School of Mathematics and Statistics Beijing Institute of Technology（北京理工大学数学与统计学院）；Academy of Mathematics and Systems Science Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Max-Planck-Institut für Mathematik（马克斯·普朗克数学研究所）
The Hecke-Clifford superalgebra is a super-analogue of the Iwahori-Hecke algebra of type A. The classification of its simple modules is done by Brundan, Kleshchev and Tsuchioka using a method of categorification of affine Lie algebras. In this paper, we introduce another way to produce its simple modules with a generalized theory of cellular algebras which is originally developed by Graham and Lehrer. In our construction the key is that there is a right action of the Clifford superalgebra on the super-analogue of the Specht module. With the help of the notion of the Morita context, a simple mo…
A Lie superalgebra is called quasi-Frobenius if it admits a closed anti-symmetric non-degenerate bilinear form. We study the notion of double extensions of quasi-Frobenius Lie superalgebra when the form is either orthosymplectic or periplectic. We show that every quasi-Frobenius Lie superalgebra that satisfies certain conditions can be obtained as a double extension of a smaller quasi-Frobenius Lie superalgebra. We classify all 4-dimensional quasi-Frobenius Lie superalgebras, and show that such Lie superalgebras must be solvable. We study the notion of $T^*$-extensions (or Lagrangian extension…
The purpose of this paper is to study Hom-Lie superalgebras, that is a superspace with a bracket for which the superJacobi identity is twisted by a homomorphism. This class is a particular case of $Γ$-graded quasi-Lie algebras introduced by Larsson and Silvestrov. In this paper, we characterize Hom-Lie admissible superalgebras and provide a construction theorem from which we derive a one parameter family of Hom-Lie superalgebras deforming the orthosymplectic Lie superalgebra. Also, we prove a $\mathbb{Z}_2$-graded version of a Hartwig-Larsson-Silvestrov Theorem which leads us to a construction…
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.
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…
We introduce the notion of $\imath$Schur superalgebra, which can be regarded as a type B/C counterpart of the $q$-Schur superalgebra (of type A) formulated as centralizer algebras of certain signed $q$-permutation modules over Hecke algebras. Some multiplication formulas for $\imath$Schur superalgebra are obtained to construct their canonical bases. Furthermore, we established an isomorphism theorem between the $\imath$Scuhr superalgebras and the $q$-Schur superalgebras of type A, which helps us derive a semisimplicity criteria of the $\imath$Schur superalgebras.
We define a natural basis for the algebra of $\frak{gosp}(1|2n)$-invariant differential operators on the affine superspace $\mathbb{C}^{1|2n}$. We prove that these operators lie in the image of the centre of the enveloping algebra of $\frak{gosp}(1|2n)$. Using this result, we compute explicit formulas for the eigenvalues of these operators on irreducible summands of $\mathcal{P}(\mathbb{C}^{1|2n})$. This settles the Capelli eigenvalue problem for orthosymplectic Lie superalgebras in the cases that were not addressed in Sahi-Salmasian-Serganova. Our main technique relies on an explicit calculat…
