On right coideal subalgebras

Integrals in left coideal subalgebras and group-like projections

We develop a theory of right group-like projections in Hopf algebras linking them with the theory of left coideal subalgebras with two sided counital integrals. Every right group-like projection is associated with a left coideal subalgebra, maximal among the ones containing the given group-like projection as an integral, and we show that that subalgebra is finite dimensional. We observe that in a semisimple Hopf algebra $H$ every left coideal subalgebra has an integral and we prove a 1-1 correspondence between right group-like projections and left coideal subalgebras of $H$. We provide a numbe…

On coideal subalgebras of cocentral Kac algebras and a generalization of Wall's conjecture

It shown that any coideal subalgebra of a finite dimensional Hopf algebra is a cyclic module over the dual Hopf algebra. Using this we describe all coideal subalgebras of a cocentral abelian extension of Hopf algebras extending some results from [4].

具有双模条件期望的余理想子代数上Hopf代数的忠实平坦性

Faithful flatness of Hopf algebras over coideal subalgebras with a bimodule conditional expectation

本文直接自足地证明了：若Hopf代数H的右余理想子代数A作为A-双模是H的直和项，则H作为左、右A-模忠实平坦。

Homogeneous right coideal subalgebras of quantized enveloping algebras

For a quantized enveloping algebra of a complex semisimple Lie algebra with deformation parameter not a root of unity, we classify all homogeneous right coideal subalgebras. Any such right coideal subalgebra is determined uniquely by a triple consisting of two elements of the Weyl group and a subset of the set of simple roots satisfying some natural conditions. The essential ingredients of the proof are the Lusztig automorphisms and the classification of homogeneous right coideal subalgebras of the Borel Hopf subalgebras of quantized enveloping algebras obtained previously by H.-J. Schneider a…

Right coideal subalgebras of quantized universal enveloping algebras of type G2

In this paper we describe the right coideal subalgebras containing all group-like elements of the two-parameter quantum groups Uq(g) and uq(g), where g is a simple Lie algebra of type G2. As a consequence, we determine that there are precisely 60 different right coideal subalgebras containing all group-like elements.

On right coideal subalgebras of quantum groups

Right coideal subalgebras are interesting substructures of Hopf algebras such as quantum groups. Examples of right coideal subalgebras are the quantum Borel part as well as quantum symmetric pairs. Classifying right coideal subalgebras is a difficult question with notable results by Schneider, Heckenberger and Kolb. After reviewing these results, as main result we prove that an arbitrary right coideal subalgebras has a particularly nice set of generators. This allows in principle to specify the set of right coideal subalgebras in a given case. As application we determine right coideal subalgeb…

Triangular decomposition of right coideal subalgebras

Let $\mathfrak g$ be a Kac-Moody algebra. We show that every homogeneous right coideal subalgebra $U$ of the multiparameter version of the quantized universal enveloping algebra $U_q(\mathfrak{g}),$ $q^m\neq 1$ containing all group-like elements has a triangular decomposition $U=U^-\otimes_{{\bf k}[F]} {\bf k}[H] \otimes_{{\bf k}[G]} U^+$, where $U^-$ and $ U^+$ are right coideal subalgebras of negative and positive quantum Borel subalgebras. However if $ U_1$ and $ U_2$ are arbitrary right coideal subalgebras of respectively positive and negative quantum Borel subalgebras, then the triangular…

Coideal Subalgebras and Quantum Symmetric Pairs

Coideal subalgebras of the quantized enveloping algebra are surveyed, with selected proofs included. The first half of the paper studies generators, Harish-Chandra modules, and associated quantum homogeneous spaces. The second half discusses various well known quantum coideal subalgebras and the implications of the abstract theory on these examples. The focus is on the locally finite part of the quantized enveloping algebra, analogs of enveloping algebras of nilpotent Lie subalgebras, and coideals used to form quantum symmetric pairs. The last family of examples is explored in detail. Connecti…

Nazarov-Wenzl algebras, coideal subalgebras and categorified skew Howe duality

We describe how certain cyclotomic Nazarov-Wenzl algebras occur as endomorphism rings of projective modules in a parabolic version of BGG category O of type $D$. Furthermore we study a family of subalgebras of these endomorphism rings which exhibit similar behaviour to the family of Brauer algebras even when they are not semisimple. The translation functors on this parabolic category O are studied and proven to yield a categorification of a coideal subalgebra of the general linear Lie algebra. Finally this is put into the context of categorifying skew Howe duality for these subalgebras.

Classifying (Weak) Coideal Subalgebras of Weak Hopf C*-Algebras

We develop a general approach to the problem of classification of weak coideal C*-subalgebras of weak Hopf C*-algebras. As an example, we consider weak Hopf C*-algebras and their weak coideal C*-subalgebras associated with Tambara Yamagami categories.

Right coideal subalgebras of the Borel part of a quantized enveloping algebra

For the Borel part of a quantized enveloping algebra we classify all right coideal subalgebras for which the intersection with the coradical is a Hopf algebra. The result is expressed in terms of characters of the subalgebras $U^+[w]$ of the quantized enveloping algebra, introduced by de Concini, Kac, and Procesi for any Weyl group element $w$. We explicitly determine all characters of $U^+[w]$ building on recent work by Yakimov on prime ideals of $U^+[w]$ which are invariant under a torus action. Key words: Quantum groups, coideal subalgebras

Right Coideal Subalgebras of the Quantum Borel Algebra of type G2

In this paper we describe the right coideal subalgebras containing all group-like elements of the multiparameter quantum group Uq+(g), where g is a simple Lie algebra of type G2, while the main parameter of quantization q is not a root of 1. If the multiplicative order t of q is finite, t>4, t different from 6, then the same classification remains valid for homogeneous right coideal subalgebras of the positive part uq+(g) of the multiparameter version of the small Lusztig quantum group.

The center of quantum symmetric pair coideal subalgebras

The theory of quantum symmetric pairs as developed by the second author is based on coideal subalgebras of the quantized universal enveloping algebra for a semisimple Lie algebra. This paper investigates the center of these coideal subalgebras, proving that the center is a polynomial ring. A basis of the center is given in terms of a submonoid of the dominant integral weights.

Kernels of representations and coideal subalgebras of Hopf algebras

We define left and right kernels of representations of Hopf algebras. In the case of group algebras, left and right kernels coincide and they are the usual kernels of modules. In the general case we show that these kernels coincide with the categorical left and right Hopf kernels of morphisms of Hopf algebras defined in \cite{AD}. Brauer's theorem for kernels over group algebras is generalized to Hopf algebras.

Braid group actions on coideal subalgebras of quantized enveloping algebras

We construct braid group actions on coideal subalgebras of quantized enveloping algebras which appear in the theory of quantum symmetric pairs. In particular, we construct an action of the semidirect product of Z^n and the classical braid group in n strands on the coideal subalgebra corresponding to the symmetric pair (sl_{2n}(C), sp_{2n}(C)). This proves a conjecture by Molev and Ragoucy. We expect similar actions to exist for all symmetric Lie algebras. The given actions are inspired by Lusztig's braid group action on quantized enveloping algebras and are defined explicitly on generators. Br…

8维Hopf代数的一维部分作用与右余理想子代数

One-dimensional partial actions of 8-dimensional Hopf algebras and right coideal subalgebras

本文完成8维Hopf代数一维部分作用的分类，证明其对称性，并证明有限维余半单Hopf代数的右余理想子代数均可实现为部分smash积。

尖点连通Hopf代数的余理想子代数

Coideal subalgebras of pointed and connected Hopf algebras

本文证明了在温和条件下，尖点Hopf代数及其连通分次版本中的余理想子代数具有PBW基结构，并借助Lyndon词和辫子换位子的组合方法构造性地建立了齐次余理想子代数对的结构定理。

Cartan Subalgebras for Quantum Symmetric Pair Coideals

There is renewed interest in the coideal subalgebras used to form quantum symmetric pairs because of recent discoveries showing that they play a fundamental role in the representation theory of quantized enveloping algebras. However, there is still no general theory of finite-dimensional modules for these coideals. In this paper, we establish an important step in this direction: we show that every quantum symmetric pair coideal subalgebra admits a quantum Cartan subalgebra which is a polynomial ring that specializes to its classical counterpart. The construction builds on Kostant and Sugiura's…

量子对称对余理想子代数的定义关系

Defining relations of quantum symmetric pair coideal subalgebras

本文确定了Kac-Moody型量子包络代数的量子对称对余理想子代数的定义关系，方法基于星积，结果用连续q-埃尔米特多项式和变形切比雪夫多项式表达。

准三角余子代数的莫里塔不变量

Morita invariants of quasitriangular coideal subalgebras

本文通过BC和D型辫群的表示，研究准三角余子代数表示范畴的莫里塔不变量，并给出K矩阵的分类结果。

Relative Fourier transforms and expectations on coideal subalgebras

For an algebraic compact quantum group $H$ we establish a bijection between the set of right coideal $*$-subalgebras $A\to H$ and that of left module quotient $*$-coalgebras $H\to C$. It turns out that the inclusion $A\to H$ always splits as a map of right $A$-modules and right $H$-comodules, and the resulting expectation $E:H\to A$ is positive (and lifts to a positive map on the full $C^*$ completion on $H$) if and only if $A$ is invariant under the squared antipode of $H$. The proof proceeds by Tannaka-reconstructing the coalgebra $C$ corresponding to $A\to H$ by means of a fiber functor fro…

Yetter-Drinfel'd algebras and coideals of Weak Hopf $C^*$-Algebras

We characterize braided commutative Yetter-Drinfeld $C^*$-algebras over weak Hopf $C^*$-algebras in categorical terms. Using this, we then study quotient type coideal subalgebras of a given weak Hopf $C^*$-algebra $\mathcal G$ and coideal subalgebras invariant with respect to the adjoint action of $\mathcal G$. Finally, as an example, we explicitly describe quotient type coideal subalgebras of the weak Hopf $C^*$-algebras associated with Tambara-Yamagami categories.

Canonical bases of a coideal subalgebra in $U_q(\mathfrak{sl}_2)$

We consider tensor products of finite-dimensional representations of a coideal subalgebra in $U_{q}(\mathfrak{sl}_2)$. We present an explicit expression for the dual of the canonical bases through a diagrammatic presentation. We show that the decomposition of tensor products of dual canonical bases and the action of the coideal subalgebra have integral and positive properties. As an application, we consider the eigensystem of the generator of the coideal subalgebra on the dual canonical bases. We provide all the eigenvalues and obtain an explicit expression of the eigenfunction for the largest…

Coideal subalgebras in quantum affine algebras

We introduce two subalgebras in the type A quantum affine algebra which are coideals with respect to the Hopf algebra structure. In the classical limit q -> 1 each subalgebra specializes to the enveloping algebra U(k), where k is a fixed point subalgebra of the loop algebra over gl_N with respect to a natural involution corresponding to the embedding of the orthogonal or symplectic Lie algebra into gl_N. We also give an equivalent presentation of these coideal subalgebras in terms of generators and defining relations which have the form of reflection-type equations. We provide evaluation homom…

Quasitriangular coideal subalgebras of $U_q(\mathfrak{g})$ in terms of generalized Satake diagrams

Let $\mathfrak{g}$ be a finite-dimensional semisimple complex Lie algebra and $θ$ an involutive automorphism of $\mathfrak{g}$. According to G. Letzter, S. Kolb and M. Balagović the fixed-point subalgebra $\mathfrak{k} = \mathfrak{g}^θ$ has a quantum counterpart $B$, a coideal subalgebra of the Drinfeld-Jimbo quantum group $U_q(\mathfrak{g})$ possessing a universal K-matrix $\mathcal{K}$. The objects $θ$, $\mathfrak{k}$, $B$ and $\mathcal{K}$ can all be described in terms of Satake diagrams. In the present work we extend this construction to generalized Satake diagrams, combinatorial data firs…

The algebra ${\cal A}_q$, $q-$Onsager algebras and coideal subalgebras: two open problems

A brief and elementary introduction to the subject is presented. Two open problems are given.

The maximal dimension of unital subalgebras of the matrix algebra

Using Wederburn's main theorem and a result of Gerstenhaber we prove that, over a field of characteristic zero, the maximal dimension of a proper unital subalgebra in the $n \times n$ matrix algebra is $n^2 - n + 1$ and furthermore this upper bound is attained for the so-called parabolic subalgebras. We also investigate the corresponding notion of parabolic coideals for matrix coalgebras and prove that the minimal dimension of a non-zero coideal of the matrix coalgebra ${\mathcal M}^n (k)$ is $n-1$.

Defining relations for quantum symmetric pair coideals of Kac-Moody type

Classical symmetric pairs consist of a symmetrizable Kac-Moody algebra $\mathfrak{g}$, together with its subalgebra of fixed points under an involutive automorphism of the second kind. Quantum group analogs of this construction, known as quantum symmetric pairs, replace the fixed point Lie subalgebras by one-sided coideal subalgebras of the quantized enveloping algebra $U_q(\mathfrak{g})$. We provide a complete presentation by generators and relations for these quantum symmetric pair coideal subalgebras. These relations are of inhomogeneous $q$-Serre type and are valid without restrictions on …

Branching rules for finite-dimensional $\mathcal{U}_q(\mathfrak{su}(3))$-representations with respect to a right coideal subalgebra

We consider the quantum symmetric pair $(\mathcal{U}_q(\mathfrak{su}(3)), \mathcal{B})$ where $\mathcal{B}$ is a right coideal subalgebra. We prove that all finite-dimensional irreducible representations of $\mathcal{B}$ are weight representations and are characterised by their highest weight and dimension. We show that the restriction of a finite-dimensional irreducible representation of $\mathcal{U}_q(\mathfrak{su}(3))$ to $\mathcal{B}$ decomposes multiplicity free into irreducible representations of $\mathcal{B}$. Furthermore we give explicit expressions for the highest weight vectors in th…

Character tables and normal left coideal subalgebras

We continue studying properties of semisimple Hopf algebras $H$ over algebraically closed fields of characteristic 0 resulting from their generalized character tables. We show that the generalized character table of $H$ reflect normal left coideal subalgebras of $H.$ These are the Hopf analogues of normal subgroups in the sense that they arise from Hopf quotients. We apply these ideas to prove Hopf analogues of known results in group theory. Among the rest we prove that columns of the character table are orthogonal and that all entries are algebraic integers. We analyze `semi kernels' and thei…

Matrix product solution to the reflection equation associated with a coideal subalgebra of $U_q(A^{(1)}_{n-1})$

We present a new solution to the reflection equation associated with a coideal subalgebra of $U_q(A^{(1)}_{n-1})$ in the symmetric tensor representations and their dual. Elements of the $K$ matrix are expressed by a matrix product formula involving terminating $q$-hypergeometric series in $q$-boson generators. At $q=0$, our result reproduces a known set-theoretical solution to the reflection equation connected to the crystal base theory.

关于量子对称对合子代数的可积模

Integrable modules over quantum symmetric pair coideal subalgebras

本文研究了量子对称对合子代数的可积模性质，并证明了其与Bao-Song坐标环的一致性。

Reflection equation algebras, coideal subalgebras, and their centres

Reflection equation algebras and related U_q(g)-comodule algebras appear in various constructions of quantum homogeneous spaces and can be obtained via transmutation or equivalently via twisting by a cocycle. In this paper we investigate algebraic and representation theoretic properties of such so called `covariantized' algebras, in particular concerning their centres, invariants, and characters. Generalising M. Noumi's construction of quantum symmetric pairs we define a coideal subalgebra B_f of U_q(g) for each character f of a covariantized algebra. The locally finite part F_l(U_q(g)) of U_q…

论量子群的 Borel 子代数

On Borel subalgebras of quantum groups

本文研究量子群中所有不可约表示均为一维的极大右余理想子代数，即 Borel 子代数，构造新例子并分析其 Verma 模、结构猜想及低秩分类。

CI型量子包络代数的余理想子代数的R矩阵表示

RTT presentation of coideal subalgebra of quantized enveloping algebra of type CI

本文利用R矩阵表示构造CI型量子包络代数的余理想子代数，通过A形式方法得到其PBW基，并构造相关Poisson代数及辫群作用。

Solution to the reflection equation related to the $ι$quantum group of type AII

A solution to the reflection equation associated to a coideal subalgebra of $U_q(A_{2n-1}^{(1)})$ of type AII in the symmetric tensor representations is presented. If parameters of the coideal subalgebra are suitably chosen, the $K$ matrix does not depend on the quantum parameter $q$ and still agrees with a solution in arXiv:nlin/0411044 at $q=0$.

带点辫子Hopf代数的一个分解定理

A decomposition Theorem for pointed braided Hopf algebras

本文推广了带点辫子Hopf代数的分解定理，证明在底层普通Hopf代数余半单时，其张量积分解与Yetter-Drinfeld范畴的余模结构相容。

右余理想子代数、Hopf Galois 对象与$\boldsymbol{\tau}$-Hopf 代数

Right Coideal Subalgebras, Hopf Galois Objects, and $\imath$-Hopf Algebras

本文研究右余模代数与 Hopf 双 Galois 对象、右余理想子代数的联系，针对 Chen 等人提出的 $\boldsymbol{\tau}$-Hopf 代数，得到其 Hopf 双 Galois 实现，完成分类并确定实现像的相等条件。

迭代Hopf Ore扩张的结构

The structure of iterated Hopf Ore extensions

本文研究特征零域上的迭代Hopf Ore扩张，证明其结构性质、PBW生成系继承定理，开发枚举其非平凡单侧余理想子代数的算法，并分类三维非余交换连通Hopf代数的非平凡右余理想子代数。

Reflection matrices, coideal subalgebras and generalized Satake diagrams of affine type

We present a generalization of the theory of quantum symmetric pairs as developed by Kolb and Letzter. We introduce a class of generalized Satake diagrams that give rise to (not necessarily involutive) automorphisms of the second kind of symmetrizable Kac-Moody algebras $\mathfrak{g}$. These lead to right coideal subalgebras $B_{\mathbf{c},\mathbf{s}}$ of quantized enveloping algebras $U_q(\mathfrak{g})$. In the case that $\mathfrak{g}$ is a twisted or untwisted affine Lie algebra of classical type Jimbo found intertwiners (equivariant maps) of the vector representation of $U_q(\mathfrak{g})$ …

Coideal algebras from twisted Manin triples

We propose a new approach to study coideal algebras. It is well-known that Manin triples (or equivalently Lie bi-algebra structures) are the requirement to deform Lie algebras and to obtain quantum groups. In this paper, introducing some particular automorphisms of Manin triples, we define new structures that we call Lie bi-ideal structures. A link with coisotropic subalgebras is explained. We show that their deformation provide coideal algebras. As examples, we recover from our general construction the twisted Yangians, the q-Onsager algebra and the augmented q-Onsager algebra. As an importan…

Symmetric pairs for Nichols algebras of diagonal type via star products

We construct symmetric pairs for Drinfeld doubles of pre-Nichols algebras of diagonal type and determine when they possess an Iwasawa decomposition. This extends G. Letzter's theory of quantum symmetric pairs. Our results can be uniformly applied to Kac-Moody quantum groups for a generic quantum parameter, for roots of unity in respect to both big and small quantum groups, to quantum supergroups and to exotic quantum groups of ufo type. We give a second construction of symmetric pairs for Heisenberg doubles in the above generality and prove that they always admit an Iwasawa decomposition. For …

Coideal Quantum Affine Algebra and Boundary Scattering of the Deformed Hubbard Chain

We consider boundary scattering for a semi-infinite one-dimensional deformed Hubbard chain with boundary conditions of the same type as for the Y=0 giant graviton in the AdS/CFT correspondence. We show that the recently constructed quantum affine algebra of the deformed Hubbard chain has a coideal subalgebra which is consistent with the reflection (boundary Yang-Baxter) equation. We derive the corresponding reflection matrix and furthermore show that the aforementioned algebra in the rational limit specializes to the (generalized) twisted Yangian of the Y=0 giant graviton.

余理想倍像表示的诱导，并应用于量子 $SL(2,\mathbb{R})$

Induction for representations of coideal doubles, with an application to quantum $SL(2,\mathbb{R})$

本文建立紧量子群 Hopf $*$-代数余理想 $*$-子代数倍像的表示诱导理论，并以量子 $SL(2,\mathbb{R})$ 为例，给出其正则表示的不可约分解。

Galois theory for H-extensions

We show that there exists a Galois correspondence between subalgebras of an H-comodule algebra A over a base ring R and generalised quotients of a Hopf algebra H if both A and H are flat Mittag--Leffler modules. We also provide new criteria for subalgebras and generalised quotients to be closed elements of the Galois connection constructed. We generalise the Schauenburg theory of admissible objects to this more general context. Then we consider coextensions of H-module coalgebras. We construct a Galois connection for them and we prove that H-Galois coextensions are closed. We apply the results…

余理想上的不变积分及其 Drinfeld 偶

Invariant integrals on coideals and their Drinfeld doubles

本文研究 CQG Hopf $*$-代数中稳定子余理想上拟不变积分的存在条件，将其扩张到 Drinfeld 偶余理想，证明其表示范畴的幺半结构，并应用于 Podleś 球对应的 $U_q(\mathfrak{sl}(2,\mathbb{R}))$。

量子矩阵球函数

Quantum Matrix Spherical Functions as Orthogonal Polynomials

本文提出了一种量子矩阵球函数理论，通过将经典三元组转化为量子三元组，得到具有差反射算子代数对角化的向量值正交多项式，并将其与中间Macdonald多项式联系起来。

Reflection $K$ matrices associated with an Onsager coideal of $U_p(A^{(1)}_{n-1})$, $U_p(B^{(1)}_n)$, $U_p(D^{(1)}_n)$ and $U_p(D^{(2)}_{n+1})$

We determine the intertwiners of a family of Onsager coideal subalgebras of the quantum affine algebra $U_p(A^{(1)}_{n-1})$ in the fundamental representations and $U_p(B^{(1)}_{n}), U_p(D^{(1)}_{n}), U_p(D^{(2)}_{n+1})$ in the spin representations. They reproduce the reflection $K$ matrices obtained recently by the matrix product construction connected to the three dimensional integrability. In particular the present approach provides the first proof of the reflection equation for the non type $A$ cases.

Solvability for semisimple Hopf algebras via integrals

We use integrals of left coideal subalgebras to develop Harmonic analysis for semisimple Hopf algebras. We show how $N^*,$ the space of functional on $N,$ is embedded in $H^*.$ We define a bilinear form on $N^*$ and show that irreducible $N$-characters are orthogonal with respect to that form. We then give an explicit formula for induced characters of $N$ and show how the induced characters are embedded in $R(H).$ In the second part we give an intrinsic definition for solvable semisimple Hopf algebras via left coideal subalgebras and their integrals. We show how this definition generalizes sol…

Galois Theory for H-extensions and H-coextensions

We show that there exists a Galois correspondence between subalgebras of an H-comodule algebra A over a base ring R and generalised quotients of a Hopf algebra H. We also show that Q-Galois subextensions are closed elements of the constructed Galois connection. Then we consider the theory of coextensions of H-module coalgebras. We construct Galois theory for them and we prove that H-Galois coextensions are closed. We apply the obtained results to the Hopf algebra itself and we show a simple proof that there is a bijection correspondence between right ideal coideals of H and its left coideal su…

Braid group actions for quantum symmetric pairs of type AIII/AIV

In the present paper we construct braid group actions on quantum symmetric pair coideal subalgebras of type AIII/AIV. This completes the proof of a conjecture by Kolb and Pellegrini in the case where the underlying Lie algebra is $\mathfrak{sl}_n$. The braid group actions are defined on the generators of the coideal subalgebras and the defining relations and braid relations are verified by explicit calculations.

Solitons, Boundaries, and Quantum Affine Algebras

This is a condensed write-up of a talk delivered at the Ramanujan International Symposium on Kac-Moody Lie algebras and Applications in Chennai in January 2002. The talk introduces special coideal subalgebras of quantum affine algebras which appear in physics when solitons are restricted to live on a half-line by an integrable boundary condition. We review how the quantum affine symmetry determines the soliton S-matrix in affine Toda field theory and then go on to use the unbroken coideal subalgebra on the half-line to determine the soliton reflection matrix. This gives a representation theore…

Bethe Subalgebras in Twisted Yangians

We study analogues of the Yangian of the Lie algebra $gl_N$ for the other classical Lie algebras $so_N$ and $sp_N$. We call them twisted Yangians. They are coideal subalgebras in the Yangian $Y(gl_N)$ of $gl_N$ and admit homomorphisms onto the universal enveloping algebras $U(so_N)$ and $U(sp_N)$ respectively. In every twisted Yangian we construct a family of maximal commutative subalgebras parametrized by the regular semisimple elements of the corresponding classical Lie algebra. The images in $U(so_N)$ and $U(sp_N)$ of these subalgebras are also maximal commutative.

量子超球面对

Quantum super-spherical pairs

本文在一般线性与正交辛量子超群中引入量子超球面对，作为求解Z2-分次反射方程矩阵的迷向子群，推广了Letzter-Kolb-Regelskis-Vlaar的量子（伪）对称对。