Dimitar Grantcharov, Ji Hye Jung, Seok-Jin Kang, Masaki Kashiwara, Myungho Kim
In this paper, we develop the crystal basis theory for the quantum queer superalgebra $\Uq$. We define the notion of crystal bases, describe the tensor product rule, and present the existence and uniqueness of crystal bases for finite-dimensional $\Uq$-modules in the category $\mathcal{O}_{int}^{\ge 0}$.
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The classical invariant theory for the queer Lie superalgebra is an investigation of the $\mathrm{U}(\mathfrak{q}_n)$-invariant sub-superalgebra of the symmetric superalgebra $\mathrm{Sym}(V^{\oplus r}\oplus V^{*\oplus s})$ for $V=\mathbb{C}^{n|n}$. We establish the first fundamental theorem of invariant theory for the quantum queer superalgebra $\mathrm{U}_q(\mathfrak{q}_n)$. The key ingredient is a quantum analog $\mathcal{O}_{r,s}$ of the symmetric superalgebra $\mathrm{Sym}(V^{\oplus r}\oplus V^{*\oplus s})$ that is created as a braided tensor product of a quantization $\mathsf{A}_{r,n}$ o…
Associated to the two types of finite dimensional simple superalgebras, there are the general linear Lie superalgebra and the queer Lie superalgebra. The universal enveloping algebras of these Lie superalgebras act on the tensor spaces of the natural representations and, thus, define certain finite dimensional quotients, the Schur superalgebras and the queer Schur superalgebra. In this paper, we introduce the quantum analogue of the queer Schur superalgebra and investigate the presentation problem for both the queer Schur superalgebra and its quantum analogue.
In this paper, we investigate the structure of highest weight modules over the quantum queer superalgebra $U_q(q(n))$. The key ingredients are the triangular decomposition of $U_q(q(n))$ and the classification of finite dimensional irreducible modules over quantum Clifford superalgebras. The main results we prove are the classical limit theorem and the complete reducibility theorem for $U_q(q(n))$-modules in the category $O_q^{\geq 0}$.
The Regular Representation of the twisted queer $q$-Schur Superalgebra
本文研究量子 queer 超代数及其扭 queer $q$-Schur 超代数的表示，证明后者正则模可分解为不可约子模直和，从而确立其半单性。
Consider the complex matrix Lie superalgebra $gl_{N|N}$ with the standard generators $E_{ij}$ where $i,j=-N,...,-1,1,...,N$. Define an involutive automorphism $η$ of $gl_{N|N}$ by $η(E_{ij})=E_{-i,-j}$. The queer Lie superalgebra $q_N$ is the fixed point subalgebra in $gl_{N|N}$ relative to the automorphism $η$. Consider the twisted polynomial current Lie superalgebra $g=\{X(t)\in\gl_{N|N}[t]:η(X(t))=X(-t)\}$. The enveloping algebra $U(g)$ of the Lie superalgebra $g$ has a deformation, called the Yangian of $q_N$. For each $M=1,2,...$ denote by $A_N^M$ the centralizer of $q_M\subset q_{N+M}$ i…
In this paper, we give an explicit combinatorial realization of the crystal B(λ) for an irreducible highest weight U_q(q(n))-module V(λ) in terms of semistandard decomposition tableaux. We present an insertion scheme for semistandard decomposition tableaux and give algorithms of decomposing the tensor product of q(n)-crystals. Consequently, we obtain explicit combinatorial descriptions of the shifted Littlewood-Richardson coefficients.
Central elements and evaluation map for the quantum queer superalgebras
本文研究量子 queer 超代数及其仿射版本的 R-矩阵表示，推导交叉对称关系并构造中心元素，同时建立从仿射代数到普通代数的同态映射。
Take the matrix Lie superalgebra $gl_{N|N}$ with the standard generators $E_{ij}$ where $i,j=-N,...,-1,1,...,N$. Define an involutive automorphism of $gl_{N|N}$ by sending $E_{ij}$ to $E_{-i,-j}$. Then the corresponding twisted subalgebra $g$ in the polynomial current Lie superalgebra $gl_{N|N}[u]$, has a natural Lie co-superalgebra structure. Here we quantise the universal enveloping algebra $U(g)$ as a co-Poisson Hopf superalgebra. For the quantised algebra we give a description of the centre, and construct the double in the sense of Drinfeld. We also construct a class of irreducible represe…
Closed formulae are constructed for the characters and dimensions of the finite dimensional simple modules of the queer Lie superalgebra q(n). This is achieved by refining Brundan's algorithm for computing simple q(n)-characters.
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…
It is shown that the direct limit of the semistandard decomposition tableau model for polynomial representations of the queer Lie superalgebra exists, which is believed to be the crystal for the upper half of the corresponding quantum group. An extension of this model to describe the direct limit combinatorially is given. Furthermore, it is shown that the polynomial representations may be recovered from the limit in most cases.
Drinfeld Yangian of a queer Lie superalgebra is defined as the quantization of a Lie bisuperelgebra of twisted polynomial currents. An analogue of the new system of generators of Drinfeld is being constructed. It is proved for the partial case Lie superalgebra $sq_1$ that this so defined Yangian and the Yangian, introduced earlier by M. Nazarov using the Faddeev-Reshetikhin-Takhtadzhjan approach, are isomorphic.
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 $\mathbb{Z}$.
Braid Group Action and Quantum Queer Superalgebra
本文通过构造辫群在泛包络超代数和量子酷儿超代数上的显式作用，重新定义根向量并给出显式表达式，从而得到包含根向量乘积的 PBW 型基。
Quantum queer superalgebra and its integral form
该研究利用辫子群作用定义量子奇异超代数的量子根向量，给出交换关系与 Lusztig 整形式的 PBW 型基，从而获得其显式表示并揭示单位根情形的结构。
The modular representation theory of the queer Lie superalgebra q(n) over characteristic p>2 is developed. We obtain a criterion for the irreducibility of baby Verma modules with semisimple p-characters and a criterion for the semisimplicity of the corresponding reduced enveloping algebras. A (2p)-power divisibility of dimensions of q(n)-modules with nilpotent p-characters is established. The representation theory of q(2) is treated in detail. We formulate a Morita super-equivalence conjecture for q(n) with general p-characters which is verified for n=2.
A solution of the associative Yang-Baxter equation related to the queer Lie superalgebra
本文提出了与queer Lie超代数相关的结合Yang-Baxter方程的一个三角解，并证明该解满足量子Yang-Baxter方程。
By using certain quantum differential operators, we construct a super representation for the quantum queer supergroup U_v(q_n). The underlying space of this representation is a deformed polynomial superalgebra in 2n^2 variables whose homogeneous components can be used as the underlying spaces of queer q-Schur superalgebras. We then extend the representation to its formal power series algebra which contains a (super) submodule isomorphic to the regular representation of U_v(q_n). A monomial basis M for U_v(q_n) plays a key role in proving the isomorphism. In this way, we may present the quantum…
Braided tensor products and polynomial invariants for the quantum queer superalgebra
本文构造量子奇异李超代数的超对称代数量子类似物，利用显式辫算子定义辫张量积并证明相关同构，最终得到其不变量的一组生成元。
Some multiplication formulas in queer $q$-Schur superalgebras
本文推导了 queer q-Schur 超代数标准基元素与偶、奇生成元的乘法公式，并利用 Hecke--Clifford 代数中的交换公式解决技术难点，为构造量子 queer 超群新实现及单位根处的 Schur--Olshanski 对偶应用奠定基础。
Constructing the quantum queer supergroup using Hecke-Clifford superalgebras
本文通过标准化queer v-Schur超代数的自然基并引入长元素取极限的方法，显式构造了从量子queer超群到queer q-Schur超代数的超代数同态，并给出了量子queer超群的一种新构造。
Approaching quantum queer supergroups using finite dimensional superalgebras (Preliminary version)
本文通过有限维queer q-Schur超代数将BLM的代数逼近方法推广到量子queer超群，建立了其新的代数实现。
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。
Results of Morse and Schilling show that the set of increasing factorizations of reduced words for a permutation is naturally a crystal for the general linear Lie algebra. Hiroshima has recently constructed two superalgebra analogues of such crystals. Specifically, Hiroshima has shown that the sets of increasing factorizations of involution words and fpf-involution words for a self-inverse permutation are each crystals for the queer Lie superalgebra. In this paper, we prove that these crystals are normal and identify their connected components. To accomplish this, we study two insertion algori…
We provide a characterization of the crystal bases for the quantum queer superalgebra recently introduced by Grantcharov et al.. This characterization is a combination of local queer axioms generalizing Stembridge's local axioms for crystal bases for simply-laced root systems, which were recently introduced by Assaf and Oguz, with further axioms and a new graph $G$ characterizing the relations of the type $A$ components of the queer crystal. We provide a counterexample to Assaf's and Oguz' conjecture that the local queer axioms uniquely characterize the queer supercrystal. We obtain a combinat…
We apply the technique of affine Hecke algebras to the invariant theory of the "queer" Lie superalgebra $q_N$. We give explicit formulas for the elements of a distinguished basis in the centre of $U(q_N)$, determined by "vanishing" conditions. We evaluate the corresponding differential operators on $C^{N|N}\otimes C^{M|M}$. In particular, we obtain an analogue for $q_N$ of the classical Capelli identity.
The quantum isomeric supercategory
本研究引入量子等距超范畴与量子仿射等距超范畴，作为HOMFLYPT纽结范畴及其仿射化的等距类似物，为量子等距超代数的表示论研究提供新的范畴工具。
Crystal bases of parabolic Verma modules over the quantum orthosymplectic superalgebras
研究显示，有理化正交辛超代数上抛物型Verma模存在唯一的晶基，该晶基由一般线性李超代数的多项式表示的q-类比诱导而来。
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…
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。
In the framework of canonical and dual canonical bases of Fock spaces, Brundan in 2003 formulated a Kazhdan-Lusztig type conjecture for the characters of the irreducible and tilting modules in the BGG category for the general linear Lie superalgebra for the first time. In this paper, we prove Brundan's conjecture and its variants associated to all Borel subalgebras in full generality.
Crystal base of the negative half of quantum orthosymplectic superalgebra
本研究构造了量子正交代数超代数负半部分的晶体基，给出其与q-振荡器表示晶体的嵌入的组合描述，借助正交代数型Burge对应新实现刻画了嵌入的复合结构。
Gelfand-Tsetlin bases of representations for super Yangian and quantum affine superalgebra
本文通过构造超 Yangian 模的 Gelfand-Tsetlin 基并给出 Drinfeld 生成元的显式作用，重新证明其不可约性，同时刻画了有限维不可约驯顺 Y(gl(1|1))-模恰为薄模，并将相应结论推广到 A 型量子仿射超代数。
Representations of quantum toroidal superalgebras and plane $\mathbf{s}$-partitions
本文构造了关联$\frak{gl}_{m|n}$李超代数与宇称$\text{s}$的量子环面超代数$\frak{E}_\text{s}$的Fock模与MacMahon模，明确其基的标记方式与生成元作用的Pieri型公式，并研究了对应特征标。
We develop a crystal base theory for the general linear Lie superalgebra $gl(m,n)$. We prove that any irreducible $U_q(gl(m,n))$-module in some category has a crystal base, and prove that its associated crystal base is parameterized by semistandard tableaux.
We introduce the notion of quantum Schur (or $q$-Schur) superalgebras. These algebras share certain nice properties with $q$-Schur algebras such as base change property, existence of canonical $\mathbb Z[v,v^{-1}]$-bases, and the duality relation with quantum matrix superalgebra $\sA(m|n)$. We also construct a cellular $\mathbb Q(\up)$-basis and determine its associated cells, called super-cells, in terms of a Robinson--Schensted--Knuth super-correspondence. In this way, we classify all irreducible representations over $\mathbb Q(\up)$ via super-cell modules.
We introduce the notion of a crystal base of a finite dimensional q-deformed Kac module over the quantum superalgebra $U_q(\gl(m|n))$, and prove its existence and uniqueness. In particular, we obtain the crystal base of a finite dimensional irreducible $U_q(\gl(m|n))$-module with typical highest weight. We also show that the crystal base of a q-deformed Kac module is compatible with that of its irreducible quotient $V(λ)$ given by Benkart, Kang and Kashiwara when $V(λ)$ is an irreducible polynomial representation.
We study representations of the quantum affine superalgebra associated with a general linear Lie superalgebra. In the spirit of Hernandez-Jimbo, we construct inductive systems of Kirillov-Reshetikhin modules based on a cyclicity result that we established previously on tensor products of these modules, and realize their inductive limits as modules over its Borel subalgebra, the so-called $q$-Yangian. A new generic asymptotic limit of the same inductive systems is proposed, resulting in modules over the full quantum affine superalgebra. We derive generalized Baxter's relations in the sense of F…
Yangian Double $DY(A(m,n))$ of Lie Superalgebra $A(m,n)$ is described in terms of generators and defining relations. It is proved triangular decomposition for Yangian $Y(A(m,n))$ and its quantum double $DY(A(m,n))$ as a corollary of PBW theorem. It is introduced the normally ordered bases in Yangian and its dual Hopf superalgebra in quantum double. It is calculated the pairing formulas between the elements of its bases. It is received the formula of Universal $R$-matrix of Yangian Double. The formula for Universal $R$-matrix of Yangian, which was introduced by V. Drinfel'd, is received also.
We obtain Drinfeld second realization of the quantum affine superalgebras associated with the affine Lie superalgebra $D^{(1)}(2,1;x)$. Our results are analogous to those obtained by Beck for the quantum affine algebras. Beck's analysis uses heavily the (extended) affine Weyl groups of the affine Lie algebras. In our approach the structures are based on a Weyl groupoid.
Let $V(λ)$ be the irreducible lowest weight $U_q(D(N,1))$-module with lowest weight $λ$. Assume $λ= n_0ω_0-\sum_{i=0}^{N}n_iω_i$, where $ω_0$ is the fundamental weight corresponding to the unique odd coroot $h_0$, and $n_i$ are positive integers. $V(λ)$ is called typical if $n_0 \geq 0$. In this paper, we construct polarizable crystal bases of $V(λ)$ in the category ${\cal O}_{int}$, which is a class of integrable modules. We also describe the decomposition of the tensor product of typical representations into irreducible ones, using the generalized Littlewood-Richardson rule for $U_q(D(N))$. …
Let $\mathcal{O}^{int}_q(m|n)$ be a semisimple tensor category of modules over a quantum ortho-symplectic superalgebra of type $B, C, D$ introduced in the author's previous work. It is a natural counterpart of the category of finitely dominated integrable modules over a quantum group of type $B, C, D$ from a viewpoint of super duality. Continuing the previous work on type $B$ and $C$, we classify the irreducible modules in $\mathcal{O}^{int}_q(m|n)$, and prove the existence and uniqueness of their crystal bases in case of type $D$. A new combinatorial model of classical crystals of type $D$ is…
We obtain Drinfel'd's realization of quantum affine superalgebra $U_q\hat{(gl(1|1))}$ based on the super version of RS construction method and Gauss decomposition.
Some introductory concepts and basic definitions of the Lie superalgebras and their quantum deformations are exposed. Especially the induced representation methods in both cases are described. Based on the Kac representation theory we have succeeded in constructing representations of several higher rank superalgebras. When representations of quantum superalgebras are concerned, we have developed a method which can be applied not only to the one-parametric quantum deformations but also to the multi-parametric ones. As an intermediate step the Gel'fand-Zetlin basis description is extended to the…
New solutions of the quantum Yang-Baxter equation, depending in general on three arbitrary parameters, are written down. They are based on the root of unity representations of the quantum orthosymplectic superalgebra \\U, which were found recently. Representations of the braid group $B_N$ are defined within any $N^{th}$ tensorial power of root of 1 \\U modules.
A representation of the quantum superalgebra Uq(sl(M+1|N+1)) is constructed based on the q-differential operators acting on the coherent states parameterized by coordinates. These coordinates correspond to the local ones of the flag manifold. This realization provides us with a guide to construct the free field realization for the quantum affine superalgebra Uq^(sl(M+1|N+1)) at arbitrary level.
In this paper, we study transition matrices of PBW bases of the nilpotent subalgebra of quantum superalgebras associated with all possible Dynkin diagrams of type A and B in the case of rank 2 and 3, and examine relationships with three-dimensional (3D) integrability. We obtain new solutions to the Zamolodchikov tetrahedron equation via type A and the 3D reflection equation via type B, where the latter equation was proposed by Isaev and Kulish as a 3D analog of the reflection equation of Cherednik. As a by-product of our approach, the Bazhanov-Sergeev solution to the Zamolodchikov tetrahedron …
We examine the two parameter deformed superalgebra $U_{qs}(sl(1|2))$ and use the results in the construction of quantum chain Hamiltonians. This study is done both in the framework of the Serre presentation and in the $R$-matrix scheme of Faddeev, Reshetikhin and Takhtajan (FRT). We show that there exists an infinite number of Casimir operators, indexed by integers $p > 1$ in the undeformed case and by $p \in Z$ in the deformed case, which obey quadratic relations. The construction of the dual superalgebra of functions on $SL_{qs}(1|2)$ is also given and higher tensor product representations a…
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