An Integral PBW Basis of the Quantum Affine Algebra of Type A$_{2}^{(2)}$

单项基与量子仿射代数的规范基

Monomial bases and canonical bases for quantum affine algebras

本文研究了量子仿射代数中单项基与规范基的构造方法，提出了一种基于PBW基的简单算法来计算规范基，并探讨了由此得到的规范基与Lusztig利用有向图几何构造的规范基之间的关系。

Representations of Khovanov-Lauda-Rouquier algebras III: Symmetric Affine Type

We develop the homological theory of KLR algebras of symmetric affine type. For each PBW basis, a family of standard modules is constructed which categorifies the PBW basis.

PBW and duality theorems for quantum groups and quantum current algebras

We give proofs of the PBW and duality theorems for the quantum Kac-Moody algebras and quantum current algebras, relying on Lie bialgebra duality. We also show that the classical limit of the quantum current algebras associated with an untwisted affine Cartan matrix is the enveloping algebra of a quotient of the corresponding toroidal algebra; this quotient is trivial in all cases except the $A_1^{(1)}$ case.

PBW theoretic approach to the module category of quantum affine algebras

Let $U_q'(\mathfrak{g})$ be a quantum affine algebra of untwisted affine ADE type and let $\mathcal{C}^0_{\mathfrak{g}}$ be Hernandez-Leclerc's category. For a duality datum $\mathcal{D}$ in $\mathcal{C}^0_{\mathfrak{g}}$, we denote by $\mathcal{F}_{\mathcal{D}}$ the quantum affine Weyl-Schur duality functor. We give sufficient conditions for a duality datum $\mathcal{D}$ to provide the functor $\mathcal{F}_{\mathcal{D}}$ sending simple modules to simple modules. Then we introduce the notion of cuspidal modules in $\mathcal{C}^0_{\mathfrak{g}}$, and show that all simple modules in $\mathcal{C}…

Representations of tame quivers and affine canonical bases

An integral PBW-basis of type $A_1^{(1)}$ has been constructed by Zhang [Z] and Chen [C] using the Auslander-Reiten quiver of the Kronecker quiver. We associate a geometric order to elements in this basis following an idea of Lusztig [L1] in the case of finite type. This leads to an algebraic realization of a bar-invariant basis of $\uq2$. For any affine symmetric type, we obtain an integral PBW-basis of the generic composition algebra, by using an algebraic construction of the integral basis for a tube in [DDX], an embedding of the module category of the Kronecker quiver into the module categ…

A PBW basis for Lusztig's form of untwisted affine quantum groups

Let $ \mathfrak{g} $ be an untwisted affine Kac-Moody algebra over the field $ K \, $, and let $ U_q(\mathfrak{g}) $ be the associated quantum enveloping algebra; let $ \mathfrak{U}_q(g) $ be the Lusztig's integer form of $ U_q(\mathfrak{g}) \, $, generated by $ q $-divided powers of Chevalley generators over a suitable subring $ R $ of $ K(q) \, $. We prove a Poincaré-Birkhoff-Witt like theorem for $ \mathfrak{U}_q(\mathfrak{g}) \, $, yielding a basis over $ R $ made of ordered products of $ q $-divided powers of suitable quantum root vectors.

利用Catalan词与$q$-shuffle代数描述$U_q(\widehat{\mathfrak{sl}}_2)$正部的Beck PBW基

Using Catalan words and a $q$-shuffle algebra to describe the Beck PBW basis for the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$

本文利用Catalan词和$q$-shuffle代数，以闭式表达量子包络代数正部的Beck PBW基，并探讨其与Damiani PBW基及交替PBW基的关系。

具有PBW基的双二次代数的微分光滑性

Differential smoothness of bi-quadratic algebras with PBW basis

该研究聚焦于具有PBW基的双二次代数，开展其微分光滑性相关的探究工作。

Rosso-Yamane Theorem on PBW basis of $U_q(A_N)$

Let $U_q(A_N)$ be the Drinfeld-Jimbo quantum group of type $A_N$. In this paper, by using Groebner-Shirshov bases, we give a simple (but not short) proof of the Rosso-Yamane Theorem on PBW basis of $U_q(A_N)$.

Diagram automorphisms and canonical bases for quantum affine algebras

Let ${\mathbf U}^-_q$ be the negative part of the quantum enveloping algebra associated to a simply laced Kac-Moody Lie algebra ${\mathfrak g}$, and $\underline{\mathbf U}^-_q$ the algebra corresponding to the fixed point subalgebra of ${\mathfrak g}$ obtained from a diagram automorphism $σ$ on ${\mathfrak g}$. Let ${\mathbf B}^σ$ be the set of $σ$-fixed elements in the canonical basis of ${\mathbf U}_q^-$, and $\underline{\mathbf B}$ the canonical basis of $\underline{\mathbf U}_q^-$. Lusztig proved that there exists a canonical bijection ${\mathbf B}^σ \simeq \underline{\mathbf B}$ based on …

具有PBW基的三生成元双二次代数的光滑几何

Smooth geometry of bi-quadratic algebras on three generators with PBW basis

本文研究了具有PBW基的三生成元双二次代数的微分光滑性，探讨了其光滑几何性质。

Two-parameter Quantum Groups of Exceptional Type E-Series and Convex PBW Type Basis

The presentation of two-parameter quantum groups of type E-series in the sense of Benkart-Witherspoon [BW1] is given, which has a Drinfel'd quantum double structure. The universal $R$-matrix and a convex PBW-type basis are described for type E_6 (as a sample), and the conditions of those isomorphisms from these quantum groups into the one-parameter quantum doubles are discussed.

PBW theory for quantum affine algebras

Let $U_q'(\mathfrak{g})$ be a quantum affine algebra of arbitrary type and let $\mathcal{C}_{\mathfrak{g}}$ be Hernandez-Leclerc's category. We can associate the quantum affine Schur-Weyl duality functor $F_D$ to a duality datum $D$ in $\mathcal{C}_{\mathfrak{g}}$. We introduce the notion of a strong (complete) duality datum $D$ and prove that, when $D$ is strong, the induced duality functor $F_D$ sends simple modules to simple modules and preserves the invariants $Λ$ and $Λ^\infty$ introduced by the authors. We next define the reflections $\mathcal{S}_k$ and $\mathcal{S}^{-1}_k$ acting on str…

修正量子群的PBW基

PBW bases for modified quantum groups

本文构造了有限型修正量子群的PBW基，证明其关于自然双线性型正交且典范基的PBW展开矩阵为单位三角矩阵，并基于最低权与最高权模张量积中元素序列的极限给出了任意型修正量子群的新构造。

The $q$-Onsager Algebra and the Universal Askey-Wilson Algebra

Recently Pascal Baseilhac and Stefan Kolb obtained a PBW basis for the $q$-Onsager algebra $\mathcal O_q$. They defined the PBW basis elements recursively, and it is obscure how to express them in closed form. To mitigate the difficulty, we bring in the universal Askey-Wilson algebra $Δ_q$. There is a natural algebra homomorphism $\natural \colon \mathcal O_q \to Δ_q$. We apply $\natural $ to the above PBW basis, and express the images in closed form. Our results make heavy use of the Chebyshev polynomials of the second kind.

Young tableaux, multi-segments, and PBW bases

The crystals for finite dimensional representations of sl(n+1) can be realized using Young tableaux. The infinity crystal on the other hand is naturally realized using multisegments, and there is a simple description of the embedding of each finite crystal into the infinity crystal in terms of these realizations. The infinity crystal is also parameterized by Lusztig's PBW basis with respect to any reduced expression for the longest word in the Weyl group. We give an explicit description of the unique crystal isomorphism from PBW bases to multisegments for one standard choice of reduced express…

Universal Enveloping Algebras of PBW Type

We continue our investigation of the general notion of universal enveloping algebra introduced in [A. Ardizzoni, \emph{A Milnor-Moore Type Theorem for Primitively Generated Braided Bialgebras}, J. Algebra \textbf{327} (2011), no. 1, 337--365]. Namely we study when such an algebra is of PBW type, meaning that a suitable PBW type theorem holds. We discuss the problem of finding a basis for a universal enveloping algebra of PBW type: As an application we recover the PBW basis both of an ordinary universal enveloping algebra and of a restricted enveloping algebra. We prove that a universal envelop…

A presentation by generators and relations of Nichols algebras of diagonal type and convex orders on root systems

We obtain a presentation by generators and relations of any Nichols algebra of diagonal type with finite root system. We prove that the defining ideal is finitely generated. The proof is based in Kharchenko's theory of PBW basis of Lyndon words. We prove that the lexicographic order on Lyndon words is convex for such PBW generators and so the PBW basis is orthogonal with respect to the canonical non-degenerate form associated to the Nichols algebra.

$q$-Onsager代数的交错中心扩张的一个新的PBW基

A new PBW basis for the alternating central extension of the $q$-Onsager algebra

为$q$-Onsager代数的交错中心扩张$\mathcal A_q$建立新PBW基，证明交错生成元在不同块序下也构成PBW基，乘法诱导同构证实猜想。

量子群的玻色子扩张的PBW理论

PBW theory for Bosonic extensions of quantum groups

本文为任意有限型量子群的玻色子扩张建立PBW理论，引入不变双线性型与伴随算子，构造具有PBW根向量和正交PBW基的子代数，推广量子幺正坐标环。

Convex PBW-type Lyndon Basis and Restricted Two-parameter Quantum Group of Type G_2

We construct finite-dimensional pointed Hopf algebras \mathfrak u_{r,s}(G_2) (i.e. restricted 2-parameter quantum groups) from the 2-parameter quantum group U_{r,s}(G_2) defined in \cite{HS}, which turn out to be of Drinfel'd doubles, where a crucial point is to give a detailed combinatorial construction of the convex PBW-type Lyndon basis for type G_2 in 2-parameter quantum version. After furnishing possible commutation relations among quantum root vectors, we show that the restricted quantum groups are ribbon Hopf algebras under certain conditions through determining their left and right int…

$U_{q,\bar{q}}(\ddot{\mathfrak{gl}}_n)$ 的 PBW 基

The PBW basis of $U_{q,\bar{q}}(\ddot{\mathfrak{gl}}_n)$

本文研究 A 型量子环面代数 $U_{q,\bar{q}}(\ddot{\mathfrak{gl}}_n)$ 的 PBW 基，证明其生成元具有类似 n=1 情形的交换关系，并给出该代数的新表示。

Remarks on PBW bases of Ringel-Hall algebras of cyclic quivers

In this paper, we give a recursive formula for the interesting PBW basis $E_{A}$ of composition subalgebras of Ringel-Hall algebras $\fkH_\vartri(n)$ of cyclic quivers after \cite{DengDuXiao2007generic}, and another construction of canonical bases of $\U_v^+(\wih{\fks\fkl}_n)$ from the monomial bases $m^{(A)}$ follow \cite{DuZhaomultiplication}. As an application, we will determined all the canonical basis of $\U_v^+(\wih{\fks\fkl}_2)$ associated with modules of Lowery length $\leqs3$. Finally, we will discuss the relation of canonical bases of Ringel-Hall algebras and those of affine quantum …

On right coideal subalgebras

Let $H$ be a character Hopf algebra. Every right coideal subalgebra that contains the coradical has a PBW-basis which can be extended up to a PBW-basis of $H.$

PBW-type filtration on quantum groups of type $A_n$

We will introduce an $\mathbb{N}$-filtration on the negative part of a quantum group of type $A_n$, such that the associated graded algebra is a q-commutative polynomial algebra. This filtration is given in terms of the representation theory of quivers, by realizing the quantum group as the Hall algebra of a quiver. We show that the induced associated graded module of any simple finite-dimensional module (of type 1) is isomorphic to a quotient of this polynomial algebra by a monomial ideal, and we provide a monomial basis for this associated graded module. This construction can be viewed as a …

仿射基的几何实现：Kronecker箭图情形

Geometric realization of affine bases: the Kronecker quiver case

本文从几何角度研究Kronecker箭图量子包络代数负部分中PBW基与标准基之间的转移矩阵，通过旗层复形构造PBW基元素的几何实现，并证明转移系数由交上同调复形限制到小子层的局部系统重数决定。

PBW bases for a class of braided Hopf algebras

We prove the existence of a basis of Poincare-Birkhoff-Witt type for braided Hopf algebras R generated by a braided subspace V of P(R) if the braiding on V fulfils a triangularity condition. We apply our result to pointed Hopf algebras with abelian coradical and to Nichols algebras of low dimensional simple U_q(sl_2) modules.

A combinatorial approach to quantification of Lie algebras

We propose a notion of a quantum universal enveloping algebra for an arbitrary Lie algebra defined by generators and relations which is based on the quantum Lie operation concept. This enveloping algebra has a PBW basis that admits the Kashiwara crystalization. We describe all skew primitive elements of the quantum universal enveloping algebra for the classical nilpotent algebras of the infinite series defined by the Serre relations and prove that the set of PBW-generators for each of these enveloping algebras coincides with the Lalonde-Ram basis of the ground Lie algebra with a skew commutato…

Braid Group Action and Quantum Affine Algebras

We lift the lattice of translations in the extended affine Weyl group to a braid group action on the quantum affine algebra. This action fixes the Heisenberg subalgebra pointwise. Loop like generators are found for the algebra which satisfy the relations of Drinfel$'$d's new realization. Coproduct formulas are given and a PBW type basis is constructed.

PBW property for associative universal enveloping algebras over an operad

Given a symmetric operad $\mathcal{P}$ and a $\mathcal{P}$-algebra $V$, the associative universal enveloping algebra ${\mathsf{U}_{\mathcal{P}}}$ is an associative algebra whose category of modules is isomorphic to the abelian category of $V$-modules. We study the notion of PBW property for universal enveloping algebras over an operad. In case $\mathcal{P}$ is Koszul a criterion for the PBW property is found. A necessary condition on the Hilbert series for $\mathcal{P}$ is discovered. Moreover, given any symmetric operad $\mathcal{P}$, together with a Gröbner basis $G$, a condition is given in…

An Algebraic Characterization of the Affine Canonical Basis

The canonical basis for quantized universal enveloping algebras associated to the finite--dimensional simple Lie algebras, was introduced by Lusztig. The principal technique is the explicit construction (via the braid group action) of a lattice over $\bz[q^{-1}]$. This allows the algebraic characterization of the canonical basis as a certain bar-invariant basis of $\cl$. Here we present a similar algebraic characterization of the affine canonical basis. Our construction is complicated by the need to introduce basis elements to span the ``imaginary'' subalgebra which is fixed by the affine brai…

对称型折叠扩张代数中PBW基与典范基的构造

Construction of PBW and canonical bases in the folding extension algebras for symmetrizable types

研究源于Lusztig构造的扩张代数，扩展Varagnolo和Vasserot方法构造斜群代数，证明相关过渡矩阵性质，为对称型情形下PBW基与典范基及其关系提供几何范畴化。

Hecke-Clifford algebras and spin Hecke algebras III: the trigonometric type

The notion of trigonometric spin double affine Hecke algebras (tsDaHa) and trigonometric double affine Hecke-Clifford algebras (tDaHCa) associated to classical Weyl groups are introduced. The PBW basis property is established. An algebra isomorphism relating tDaHCa to tsDaHa is obtained.

Hecke-Clifford algebras and spin Hecke algebras I: the classical affine type

Associated to the classical Weyl groups, we introduce the notion of degenerate spin affine Hecke algebras and affine Hecke-Clifford algebras. For these algebras, we establish the PBW properties, formulate the intertwiners, and describe the centers. We further develop connections of these algebras with the usual degenerate (i.e. graded) affine Hecke algebras of Lusztig by introducing a notion of degenerate covering affine Hecke algebras.

关于量子仿射超代数 $U_{v}(A(0,2)^{(4)})$ 的结构

On the structure of quantum affine superalgebra $U_{v}(A(0,2)^{(4)})$

本研究针对非单位根参数下由量子Serre关系定义的量子仿射超代数正部分 $U_{v}(A(0,2)^{(4)})^{+}$，证明其与Nichols代数同构，明确了两种定义方式的等价性，同时确定了所有根重数并给出PBW基。

Double affine Hecke algebras for the spin symmetric group

We introduce a new class (in two versions) of rational double affine Hecke algebras (DaHa) associated to the spin symmetric group. We establish the basic properties of the algebras, such as PBW and Dunkl representation, and connections to Nazarov's degenerate affine Hecke-Clifford algebra and to a new degenerate affine Hecke algebra introduced here. We formulate a Morita equivalence between the two versions of rational DaHa's. The trigonometric generalization of the above constructions is also formulated and its relation to the rational counterpart is established.

The PBW Filtration, Demazure Modules and Toroidal Current Algebras

Let $L$ be the basic (level one vacuum) representation of the affine Kac-Moody Lie algebra $\hat{\mathfrak g}$. The $m$-th space $F_m$ of the PBW filtration on $L$ is a linear span of vectors of the form $x_1... x_lv_0$, where $l\le m$, $x_i\in \hat{\mathfrak g}$ and $v_0$ is a highest weight vector of $L$. In this paper we give two descriptions of the associated graded space $L^{\rm gr}$ with respect to the PBW filtration. The "top-down" description deals with a structure of $L^{\rm gr}$ as a representation of the abelianized algebra of generating operators. We prove that the ideal of relatio…

Deformed preprojective algebras and symplectic reflection algebras for wreath products

We determine the PBW deformations of the wreath product of a symmetric group with a deformed preprojective algebra of an affine Dynkin quiver. In particular, we show that there is precisely one parameter which does not come from deformation of the preprojective algebra. We prove that the PBW deformation is Morita equivalent to a corresponding symplectic reflection algebra for wreath product.

Hecke-Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type

We introduce an odd double affine Hecke algebra (DaHa) generated by a classical Weyl group W and two skew-polynomial subalgebras of anticommuting generators. This algebra is shown to be Morita equivalent to another new DaHa which are generated by W and two polynomial-Clifford subalgebras. There is yet a third algebra containing a spin Weyl group algebra which is Morita (super)equivalent to the above two algebras. We establish the PBW properties and construct Verma-type representations via Dunkl operators for these algebras.

Hecke-Clifford algebras and spin Hecke algebras II: the rational double affine type

The notion of rational spin double affine Hecke algebras (sDaHa) and rational double affine Hecke-Clifford algebras (DaHCa) associated to classical Weyl groups are introduced. The basic properties of these algebras such as the PBW basis and Dunkl operator representations are established. An algebra isomorphism relating the rational DaHCa to the rational sDaHa is obtained. We further develop a link between the usual rational Cherednik algebra and the rational sDaHa by introducing a notion of rational covering double affine Hecke algebras.

PBW bases and KLR algebras

We generalize Lusztig's geometric construction of the PBW bases of finite quantum groups of type $\mathsf{ADE}$ under the framework of [Varagnolo-Vasserot, J. reine angew. Math. 659 (2011)]. In particular, every PBW basis of such quantum groups is proven to yield a semi-orthogonal collection in the module category of the KLR-algebras. This enables us to prove Lusztig's conjecture on the positivity of the canonical (lower global) bases in terms of the (lower) PBW bases in the $\mathsf{ADE}$ case. In addition, we verify Kashiwara's problem on the finiteness of the global dimensions of the KLR-al…

Rational Ringel-Hall algebras, Hall polynomials of affine type and Canonical bases

In this paper, the rational Ringel-Hall algebras for tame quivers are introduced and are identified with the positive part of the quantum extended Kac-Moody algebras. By using the rational Ringel-Hall algebras, we show that the existence of Hall polynomials for tame quiver algebras. The PBW bases are constructed and new classes of perverse sheaves are shown to have strong purity property. These allows us to construct the canonical bases of the positive part of the quantum extended Kac-Moody algebras.

q - 昂萨格代数的PBW基与中心化子

PBW bases and centralisers for the $q$-Onsager algebra

该研究解决了q - 昂萨格代数PBW基与中心化子的五个猜想，通过证明特定根向量给出PBW基、建立多个PBW基及确定相关子代数中心化子，运用多种论证方法得出四个单族交替多项式子代数是极大交换的结论。

仿射杨代数作为量子扭代数的极限

Affine Yangians as Limits of Quantum Toroidal Algebras

本文证明了量子扭代数与非扭曲仿射杨代数之间存在退化同构，建立了仿射杨代数的PBW基和经典极限为多项式电流李代数的结构特性。

一种量子洗牌方法用于类型 $C(2)^{(2)}$ 的量子拟叠代数及其公平呈现

A quantum shuffle approach to quantum affine super algebra of type $C(2)^{(2)}$ and its equitable presentation

本文提出利用量子洗牌方法和卡塔兰词，以闭式表达式呈现类型 $C(2)^{(2)}$ 的量子拟叠代数的两个基，并给出了其玻色化表示。

具有PBW基和GK维数的自由双代数的商Hopf代数

Quotient Hopf algebras of the free bialgebra with PBW bases and GK-dimensions

本文研究由二维余代数生成的自由双代数T及其商，揭示其与非交换Faà di Bruno双代数的关系，并在正特征下构造具有有限GK维数的商Hopf代数链，分析其同调性质与余根滤过。

A parameterization of the canonical bases of affine modified quantized enveloping algebras

For symmetrizable Kac-Moody Lie algebra $\textbf{g}$, Lusztig introduced the modified quantized enveloping algebra $\dot{\textbf{U}}(\textbf{g})$ and its canonical basis in [12]. In this paper, for finite and affine type symmetric Lie algebra $\textbf{g}$ we define a set which depend only on the root category and prove that there is a bijection between the set and the canonical basis of $\dot{\textbf{U}}(\textbf{g})$, where the root category is the $T^2$-orbit category of the derived category of Dynkin or tame quiver. Our method bases on one theorem of Lin, Xiao and Zhang in [9], which gave th…

Classification of quadratic and cubic PBW algebras on three generators

We give a complete classification of quadratic algebras A, with Hilbert series $H_A=(1-t)^{-3}$, which is the Hilbert series of commutative polynomials on 3 variables. Koszul algebras as well as algebras with quadratic Gröbner basis among them are identified. We also give a complete classification of cubic algebras A with Hilbert series $H_A=(1+t)^{-1}(1-t)^{-3}$. These two classes of algebras contain all Artin-Schelter regular algebras of global dimension 3. As far as the latter are concerned, our results extend well-known results of Artin and Schelter by providing a classification up to an a…

$t$-Analog of $q$-Characters, Bases of Quantum Cluster Algebras, and a Correction Technique

We first study a new family of graded quiver varieties together with a new $t$-deformation of the associated Grothendieck rings. This provides the geometric foundations for a joint paper by Yoshiyuki Kimura and the author. We further generalize the result of that paper to any acyclic quantum cluster algebra with arbitrary nondegenerate coefficients. In particular, we obtain the generic basis, the dual PBW basis, and the dual canonical basis. The method consists in a correction technique, which works for general quantum cluster algebras.

On the geometric realization of the inner product and canonical basis for quantum affine $\mathfrak{sl}_n$

We give a geometric interpretation of the inner product on the modified quantum group of $\hat{\mathfrak{sl}}_n$. We also give some applications of this interpretation, including a positivity result for the inner product, and a new geometric construction of the canonical basis.

扭曲$q$-Yangian与仿射$\imath$量子群之间的同构：AI型

Isomorphism between twisted $q$-Yangians and affine $\imath$quantum groups: type AI

本研究利用高斯分解，建立了AI型对称对对应的扭曲$q$-Yangian与仿射$\imath$量子群的显式同构，并由此得到该类仿射$\imath$量子群的PBW型基。

量子群的结构系数

Structure coefficients for quantum groups

本研究基于带自同构箭图的Hall代数，给出有限型量子群结构系数的几何实现，提供Hall多项式存在性的新证明，并推广PBW基bar对合的相关表达式。

Drinfeld Basis And a Nonclassical Free Boson Representation of Twisted Quantum Affine Superalgebra $U_q[osp(2|2)^{(2)}]$}

We derive from the super RS algebra the Drinfeld basis of the twisted quantum affine superalgebra $U_q[osp(2|2)^{(2)}]$ by means of the Gauss decomposition technique. We explicitly construct a nonclassical level-one representation of $U_q[osp(2|2)^{(2)}]$ in terms of two $q$-deformed free boson fields.

Tetrahedron and 3D reflection equation from PBW bases of the nilpotent subalgebra of quantum superalgebras

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 …