A. Davydov
We classify coproducts on matrix algebra in terms of solutions to some modification of pentagon equation. The construction of Baaj and Skandalis describing finite dimensional unitary solutions of pentagon equation is extended to the non-unitary case. We establish the relation between Hopf-Galois algebras and solutions to modified pentagon equation.
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The direct sum ${\cal O}_{*}$ of all Cuntz algebras has a non-cocommutative comultiplication $Δ_φ$ such that there exists no antipode of any dense subbialgebra of the C$^{*}$-bialgebra $({\cal O}_{*},Δ_φ)$. From states equations of ${\cal O}_{*}$ with respect to the tensor product, we construct an operator $W$ for $({\cal O}_{*},Δ_φ)$ such that $W^{*}$ is an isometry, $W(x\otimes I)W^{*}=Δ_φ(x)$ for each $x\in {\cal O}_{*}$ and $W$ satisfies the pentagon equation.
New types of bialgebras arising from the Hopf equation (pentagonal equation) are introduced and studied. They will play from the Hopf equation the same role as the co-quasitriangular do from the quantum Yang Baxter equation.
For a C$^{*}$-bialgebra $A$ with a comultiplication $Δ$, a universal $R$-matrix of $(A,Δ)$ is defined as a unitary element in the multiplier algebra $M(A\otimes A)$ of $A\otimes A$ which is an intertwiner between $Δ$ and its opposite comultiplication $Δ^{op}$. We show that there exists no universal $R$-matrix for some C$^{*}$-bialgebras.
The quantum bialgebra related to the Baxter's eight-vertex R-matrix is found as a quantum deformation of the Lie algebra of sl(2)-valued automorphic functions on a complex torus.
Let $M_{*}({\bf C})$ denote the C$^{*}$-algebra defined as the direct sum of all matrix algebras $\{M_{n}({\bf C}):n\geq 1\}$. It is known that $M_{*}({\bf C})$ has a non-cocommutative comultiplication $Δ_φ$. We show that the C$^{*}$-bialgebra $(M_{*}({\bf C}),Δ_φ)$ has a universal $R$-matrix $R$ such that the quasi-cocommutative C$^{*}$-bialgebra $(M_{*}({\bf C}),Δ_φ,R)$ is triangular.
The dimension of the third homogeneous component of a matrix quantum bialgebra, determined by pair of quantum spaces, is calculated. The Poincaré series of some deformations of $GL(n)$ is calculated. A new deformation of $GL(3)$ with the correct dimension is given.
Bialgebras associated to Yang-Baxter operators satisfying the Hecke equation, are considered. It is shown that they are Koszul algebras. Their Poincare' series are calculated via the Poincare' series of the corresponding quantum spaces.
In the classification of Hietarinta, three triangular $4\times 4$ $R$-matrices lead, via the FRT formalism, to matrix bialgebras which are not deformations of the trivial one. In this paper, we find the bialgebras which are in duality with these three exotic matrix bialgebras. We note that the $L-T$ duality of FRT is not sufficient for the construction of the bialgebras in duality. We find also the quantum planes corresponding to these bialgebras both by the Wess-Zumino R-matrix method and by Manin's method.
We present the exotic bialgebras that arise from a 9x9 unitary braid matrix.
In this paper we investigate the algebraic structure of AdS/CFT in the strong-coupling limit. We propose an expression for the classical r-matrix with (deformed) u(2|2) symmetry, which leads to a quasi-triangular Lie bialgebra as the underlying symmetry algebra. On the fundamental representation our r-matrix coincides with the classical limit of the quantum R-matrix.
A new construction method of $R$-matrix is given. Let $A$ be a C$^{*}$-bialgebra with a comultiplication $Δ$. For two states $ω$ and $ψ$ of $A$ which satisfy certain conditions, we construct a unitary $R$-matrix $R(ω,ψ)$ of the C$^{*}$-bialgebra $(A,Δ)$ on the tensor product of GNS representation spaces associated with $ω$ and $ψ$. The set $\{R(ω,ψ):ω,ψ\}$ satisfies a kind of Yang-Baxter equation. Furthermore, we show a nontrivial example of such $R$-matrices for a non-quasi-cocommutative C$^{*}$-bialgebra.
Generalized splitting of algebras with application to a bialgebra structure of Leibniz algebras induced from averaging Lie bialgebras
本文扩展了operad的经典分裂概念，提出了由系数矩阵参数化的type-M pre-结构族，并将其应用于确定由平均Lie双代数诱导的Leibniz代数双代数结构。
We describe the Lie bialgebra structure on the Lie superalgebra sl(2,1) related to an r-matrix that cannot be obtained by a Belavin-Drinfeld type construction. This structure makes sl(2,1) into the Drinfeld double of a four-dimensional subalgebra.
We find the exotic matrix bialgebras which correspond to the two non-triangular nonsingular 4x4 R-matrices in the classification of Hietarinta, namely, R_{S0,3} and R_{S1,4}. We find two new exotic bialgebras S03 and S14 which are not deformations of the of the classical algebras of functions on GL(2) or GL(1|1). With this we finalize the classification of the matrix bialgebras which unital associative algebras generated by four elements. We also find the corresponding dual bialgebras of these new exotic bialgebras and study their representation theory in detail. We also discuss in detail a sp…
All possible Lie bialgebra structures on the harmonic oscillator algebra are explicitly derived and it is shown that all of them are of the coboundary type. A non-standard quantum oscillator is introduced as a quantization of a triangular Lie bialgebra, and a universal $R$-matrix linked to this new quantum algebra is presented.
This paper is a continuation of "Quantization of Lie bialgebras I-IV". The goal of this paper is to define and study the notion of a quantum vertex operator algebra in the setting of the formal deformation theory and give interesting examples of such algebras. In particular, we construct a quantum vertex operator algebra from a rational, trigonometric, or elliptic R-matrix, which is a quantum deformation of the affine vertex operator algebra. The simplest vertex operator in this algebra is the quantum current of Reshetikhin and Semenov-Tian-Shansky.
The paper deals with three topics on coquasitriangular bialgebras. A characterization of universal r-forms in terms of Yetter-Drinfeld modules is given. All universal r-forms for the coordinate Hopf algebras of the quantum groups GL_q(N), SL_q(N), O_q(N) and Sp_q(N) are described. It is well-known that the square of the antipode of a coquasitriangular Hopf algebra A with universal r-form r can be expressed by means of the linear functional f_r(a)=r(a_{(1)}\otimes S(a_{(2)})), a\in A. For the standard compact matrix groups the functional f_r is related to the Woronowicz modular character f_{-2}…
We study a possibility to define the (braided) comultiplication for the GLq(N)-covariant differential complexes on some quantum spaces. We discover such `differential bialgebras' (and Hopf algebras) on the bosonic and fermionic quantum hyperplanes (with additive coproduct) and on the braided matrix algebra BMq(N) with both multiplicative and additive coproducts. The latter case is related (for N=2) to the q-Minkowski space and q-Poincare algebra.
Non-standard quantum groups $C_R [GL(n)]$ and $C_R [SL(n)]$ are constructed for a two parameter version of the Cremmer-Gervais $R$-matrix. An epimorphism is constructed from $C_R [GL(n)]$ onto the restricted dual $U_{\bar{R}}(\frak{gl}(n-1))$ associated to a related smaller $R$-matrix of the same form. A related result is proved concerning factorizable Lie bialgebras. For any such Lie bialgebra, the dual Lie bialgebra has a canonical homomorphic image which is again factorizable.
The dual Lie bialgebra of a certain ``quasitriangular'' Lie bialgebra structure on the Heisenberg Lie algebra determines a (non-compact) Poisson--Lie group G. The compatible Poisson bracket on G is non-linear, but it can still be realized as a ``cocycle perturbation'' of the linear Poisson bracket. We construct a certain twisted group C*-algebra A, which is shown to be a strict deformation quantization of G. Motivated by the data at the Poisson (classical) level, we then construct on A its locally compact quantum group structures: comultiplication, counit, antipode and Haar weight, as well as …
This paper is a continuation of "Quantization of Lie bialgebras, III" (q-alg/9610030, revised version). In QLB-III, we introduced the Hopf algebra F(R)_\z associated to a quantum R-matrix R(z) with a spectral parameter, and a set of points \z=(z_1,...,z_n). This algebra is generated by entries of a matrix power series T_i(u), i=1,...,n,subject to Faddeev-Reshetikhin-Takhtajan type commutation relations, and is a quantization of the group GL_N[[t]]. In this paper we consider the quotient F_0(R)_\z of F(R)_\z by the relations \qdet_R(T_i)=1, where \qdet_R is the quantum determinant associated to…
Let A be a finite dimensional Hopf algebra and (H, R) a quasitriangular bialgebra. Denote by H^*_R a certain deformation of the multiplication of H^* via R. We prove that H^*_R is a quantum commutative left H\otimes H^{op cop}-module algebra. If H is the Drinfel'd double of A then H^*_R is the Heisenberg double of A. We study the relation between H^*_R and Majid's "covariantised product". We give a formula for the canonical element of the Heisenberg double of A, solution to the pentagon equation, in terms of the R-matrix of the Drinfel'd double of A. We generalize a theorem of Jiang-Hua Lu on …
We introduce a pentagon equation solver, available as part of SageMath, and use it to construct braid group representations associated to certain anyon systems. We recall the category-theoretic framework for topological quantum computation to explain how these representations describe the sets of logical gates available to an anyonic quantum computer for information processing. In doing so, we avoid venturing deep into topological or conformal quantum field theory. Instead, we present anyons abstractly as sets of labels together with a collection of data satisfying a number of axioms, includin…
All Lie bialgebra structures for the (1+1)-dimensional centrally extended Schrodinger algebra are explicitly derived and proved to be of the coboundary type. Therefore, since all of them come from a classical r-matrix, the complete family of Schrodinger Poisson-Lie groups can be deduced by means of the Sklyanin bracket. All possible embeddings of the harmonic oscillator, extended Galilei and gl(2) Lie bialgebras within the Schrodinger classification are studied. As an application, new quantum (Hopf algebra) deformations of the Schrodinger algebra, including their corresponding quantum universa…
This paper is a continuation of the series of papers "Quantization of Lie bialgebras (QLB) I-V". We show that the image of a Kac-Moody Lie bialgebra with the standard quasitriangular structure under the quantization functor defined in QLB-I,II is isomorphic to the Drinfeld-Jimbo quantization of this Lie bialgebra, with the standard quasitriangular structure. This implies that when the quantization parameter is formal, then the category O for the quantized Kac-Moody algebra is equivalent, as a braided tensor category, to the category O over the corresponding classical Kac-Moody algebra, with th…
Affine Classical Lie Bialgebras for AdS/CFT Integrability
本文将 gl(2|2) 李双代数的收缩与约化构造推广至仿射代数 sl(2)^(1)×d(2,1;α)^(1)，用以刻画 AdS/CFT 与 Hubbard 模型可积性中 r-矩阵偏离差分形式及 Lorentz boost 对称性。
1-shifted Lie bialgebras and their quantizations
本文定义并研究了1-移位李双代数及其Manin三元组，构造了1-移位度量李代数的典范量子化和横截Lagrangian子代数的量子化，并应用于圈群李代数得到Yangian的上同调移位类比。
Darboux families and the classification of real four-dimensional indecomposable coboundary Lie bialgebras
本文提出 Darboux 族概念，用于对实四维不可分解李代数上的上边缘李双代数、r-矩阵和经典 Yang-Baxter 方程解进行几何分析与自同构分类，并推广到高维情形。
Yang-Baxter bialgebras, as previously introduced by the authors, are shown to arise from a double crossproduct construction applied to the bialgebra R T T = T T R, E T = T E R, Δ(T) = T \hat{\otimes} T, Δ(E) = E \hat{\otimes} T + 1 \hat{\otimes} E and its skew dual, with R being a numerical matrix solution of the Yang-Baxter equation. It is further shown that a set of relations generalizing q-Serre ones in the Drinfeld-Jimbo algebras U_q(g) can be naturally imposed on Yang-Baxter algebras from the requirement of non-degeneracy of the pairing.
The Drinfeld double of a finite dimensional Hopf algebra is a quasi-triangular Hopf algebra with the canonical element as the universal $R$-matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant of framed links is constructed using a ribbon Hopf algebra. In that construction, a copy of the universal $R$-matrix is attached to each crossing, and invariance under the Reidemeister III move is shown by the quantum Yang-Baxter equation of the universal $R$-matrix. On the other hand, the Heisenberg double of a finite dimensional Hopf algebra has…
We define a new kind quantized enveloping algebra of a generalized Kac-Moody algebra ${\mathcal G}$ by adding a new generator $J$ satisfying $J^m=J$ for some integer $m$. We denote this algebra by $wU_q^τ({\mathcal G})$. This algebra is a weak Hopf algebra if and only if $m=2,3$. In general, it is a bialgebra, and contains a Hopf subalgebra. This Hopf subalgebra is isomorphic to the usually quantum envelope algebra $U_q({\mathcal G})$ of a generalized Kac-Moody algebra ${\mathcal G}$.
We describe the diagonal reduction algebra D(gl(n)) of the Lie algebra gl(n) in the R-matrix formalism. As a byproduct we present two families of central elements and the braided bialgebra structure of D(gl(n)).
We find and classify all bialgebras and Hopf algebras or `quantum groups' of dimension $\le 4$ over the field $\Bbb F_2=\{0,1\}$. We summarise our results as a quiver, where the vertices are the inequivalent algebras and there is an arrow for each inequivalent bialgebra or Hopf algebra built from the algebra at the source of the arrow and the dual of the algebra at the target of the arrow. There are 314 distinct bialgebras, and among them 25 Hopf algebras with at most one of these from one vertex to another. We find a unique smallest noncommutative and noncocommutative one, which is moreover s…
We report some observations concerning two well-known approaches to construction of quantum groups. Thus, starting from a bialgebra of inhomogeneous type and imposing quadratic, cubic or quartic commutation relations on a subset of its generators we come, in each case, to a q-deformed universal enveloping algebra of a certain simple Lie algebra. An interesting correlation between the order of initial commutation relations and the Cartan matrix of the resulting algebra is observed. Another example demonstrates that the bialgebra structure of sl_q(2) can be completely determined by requiring the…
Infinitesimal $\mathcal{R}$-matrices for some families of Hopf algebras
本文研究了某些著名Hopf代数家族的无穷小R矩阵的分类，基于双代数的准三角结构定义并给出了具体结果。
Universal K-matrices for quantum Kac-Moody algebras
本文提出柱形双代数与泛 K-矩阵概念，证明 Kac-Moody 型量子对称对可构造依赖广义 Satake 图的泛解，并在 $U_qL\mathfrak{sl}_2$ 有限维表示中得到带谱参数的形式解。
For any algebra two families of coloured Yang-Baxter operators are constructed, thus producing solutions to the two-parameter quantum Yang-Baxter equation. An open problem about a system of functional equations is stated. The matrix forms of these operators for two and three dimensional algebras are computed. A FRT bialgebra for one of these families is presented. Solutions for the one-parameter quantum Yang-Baxter equation are derived and a Yang-Baxter system constructed.
This article continues the study of concrete algebra-like structures in our polyadic approach, where the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions ("arity freedom principle"). In this way, generalized associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both the algebra and its underlying field can be zeroless and nonunital, the existence of the uni…
Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groups $U_q(g)$. They have the same FRT generators $l^\pm$ but a matrix braided-coproduct $\undΔL=L\und\tens L$ where $L=l^+Sl^-$, and are self-dual. As an application, the degenerate Sklyanin algebra is shown to be isomorphic to the braided matrices $BM_q(2)$; it is a braided-commutative bialgebra in a braided category…
Quivers with quantum Yang-Baxter equation and Hecke condition: Deformation of face algebras
本文引入带量子Yang-Baxter和Hecke结构的箭图，研究其路代数中由Hecke R-矩阵确定的RTT关系，证明量子矩阵代数与玫瑰箭图上的面代数同构。
Quantum upper triangular matrix algebras
基于先前工作，构造了双代数T_q(n)和尖点Hopf代数UT_q(n)，分别量子化上三角矩阵代数和可逆上三角矩阵群，并研究了n=2时的导子李代数、自同构群和低阶Hochschild上同调。
We show that if $A$ and $H$ are Hopf algebras that have equivalent tensor categories of comodules, then one can transport what we call a free Yetter-Drinfeld resolution of the counit of $A$ to the same kind of resolution for the counit of $H$, exhibiting in this way strong links between the Hochschild homologies of $A$ and $H$. This enables us to get a finite free resolution of the counit of $\mathcal B(E)$, the Hopf algebra of the bilinear form associated to an invertible matrix $E$, generalizing an ealier construction of Collins, Hartel and Thom in the orthogonal case $E=I_n$. It follows tha…
Quantum unitary group $U_{q,Θ}(3)$ for complex deformation parameters
利用Hayashi R-矩阵和FRT双代数技术，构造了非零实q和模一复形变参数θ_{ij}的紧量子群U_{q,Θ}(3)，并通过三维非交换环面的表示研究了C*-代数C(U_{q,Θ}(3))的不可约*-表示。
Free resolutions for free unitary quantum groups and universal cosovereign Hopf algebras
本研究构造了自由酉量子群与一般参数矩阵下泛共主权Hopf代数余单位的有限自由分解，结合Yetter-Drinfeld结构与相关理论计算了其上同调，并推广了粘合自由积与投射分解的适用范围。
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