quantum-group: Exact symbolic verification of R-matrix and Yang–Baxter identities for U_q(sl_2) and GL_q(2|1)

Taha Berk Terekli

Published Sep 29, 2026• 0 citations

Abstract

Explicit R-matrices of quantum groups are routinely transcribed from the literature, yet whether they satisfy an identity depends on basis order, tensor placement, the coproduct and, for superalgebras, the Koszul sign rule. quantum-group is a Python package that builds such matrices exactly with SymPy and returns every identity as an exact residual matrix, whose nonzero entries locate the convention that disagrees. It covers U_q(sl_2) modules, tensor products and the fundamental R-matrix, and the graded Yang–Baxter equation for GL_q(2|1). Applied to the thesis it was written for, it exposed an R-matrix that satisfies the Yang–Baxter equation but intertwines the opposite coproduct. Userguide of quantum-group 1.1.1 (software: https://doi.org/10.5281/zenodo.23039671; source: https://github.com/TerekliTahaBerk/quantum-groups). Prepared for submission to SciPost Physics Codebases.

DOI: 10.5281/zenodo.23040092

Related papers

Quantum algebra deforming maps, Clebsch–Gordan coefficients, coproducts, R and U matrices

Thomas Curtright, G. I. Ghandour, COSMAS K. ZACHOS

Journal of Mathematical PhysicsMar 1, 199149 citations

Quantum Group Construction of Non-standard R-matrix for Yang-Baxter Equation

Mo‐Lin Ge, Sun Changpu, Kang Xue

By taking the concept of weight conservation into account, new solutions of Yang-Baxter equation without spectral parameter are obtained through the non-standard R-matrix on the tensorial space V 1/2 ⊗ V j ( j ≠ 1/2) in terms of the irreducible representation of the quantum universal enveloping algebra SL q (2). The non-standard R-matrix R 1/2 1 is obtained in an explicit form for j = 1.

Chinese Physics LettersOct 1, 19910 citations

Some functional solutions of the Yang-Baxter equation

D. Ts. Stoyanov

arXiv:hep-th/9405179May 27, 19940 citations

Functional relations and the Yang-Baxter algebra

W. Galléas

arXiv:1312.6816Journal of Physics Conference SeriesNov 29, 20138 citations

Quantum groups

M. Ruiz-Altaba

arXiv:hep-th/9311069AIP conference proceedingsJan 1, 19940 citations

The Six-Vertex Yang-Baxter Groupoid

Daniel Bump, Slava Naprienko

arXiv:2503.05960Mar 7, 20250 citations

quantum-group: Exact symbolic verification of R-matrix and Yang–Baxter identities for U_q(sl_2) and GL_q(2|1)

Taha Berk Terekli

Published Sep 29, 2026• 0 citations

Publisher

Abstract

DOI: 10.5281/zenodo.23040092

Related papers

Quantum algebra deforming maps, Clebsch–Gordan coefficients, coproducts, R and U matrices

Thomas Curtright, G. I. Ghandour, COSMAS K. ZACHOS

Quantum algebra deforming maps explicitly define comultiplications that differ from the usual noncocommutative coproducts. Map-induced coproducts are connected to the usual ones by similarity transformations U that may be expressed either in terms of Clebsch–Gordan coefficients, or in a universal operator form. The product of two such U matrices yields the R matrix for a fixed value of the spectral parameter, which bears on the Yang–Baxterization of U as well as R. All this is explicitly illustrated for the tensor product 1/2⊗j of SU(2)q using several deforming maps whose coproducts are continuously connected by similarity transformations to form a two-parameter manifold. Some observations are made on the general structure of U and R matrices, and of coproduct manifolds, based on the solutions of hierarchies of partial difference equations. Applications of deforming maps and U matrices to the physics of spin-chains are outlined.Quantum algebra deforming maps explicitly define comultiplications that differ from the usual noncocommutative coproducts. Map-induced coproducts are connected to the usual ones by similarity transformations U that may be expressed either in terms of Clebsch–Gordan coefficients, or in a universal operator form. The product of two such U matrices yields the R matrix for a fixed value of the spectral parameter, which bears on the Yang–Baxterization of U as well as R. All this is explicitly illustrated for the tensor product 1/2⊗j of SU(2)q using several deforming maps whose coproducts are continuously connected by similarity transformations to form a two-parameter manifold. Some observations are made on the general structure of U and R matrices, and of coproduct manifolds, based on the solutions of hierarchies of partial difference equations. Applications of deforming maps and U matrices to the physics of spin-chains are outlined.

Journal of Mathematical PhysicsMar 1, 199149 citations

Quantum Group Construction of Non-standard R-matrix for Yang-Baxter Equation

Mo‐Lin Ge, Sun Changpu, Kang Xue

Chinese Physics LettersOct 1, 19910 citations

Some functional solutions of the Yang-Baxter equation

D. Ts. Stoyanov

A general functional definition of the infinite dimensional quantum $R$ -matrix satisfying the Yang-Baxter equation is given. A procedure for the extracting a finite dimensional $R$ -matrix from the general definition is demonstrated in a particular case when the group $SU(2)$ takes place.A general functional definition of the infinite dimensional quantum $R$-matrix satisfying the Yang-Baxter equation is given. A procedure for the extracting a finite dimensional $R$-matrix from the general definition is demonstrated in a particular case when the group $SU(2)$ takes place.

arXiv:hep-th/9405179May 27, 19940 citations

Functional relations and the Yang-Baxter algebra

W. Galléas

Functional equations methods are a fundamental part of the theory of Exactly Solvable Models in Statistical Mechanics and they are intimately connected with Baxter's concept of commuting transfer matrices. This concept has culminated in the celebrated Yang-Baxter equation which plays a fundamental role for the construction of quantum integrable systems and also for obtaining their exact solution. Here I shall discuss a proposal that has been put forward in the past years, in which the Yang-Baxter algebra is viewed as a source of functional equations describing quantities of physical interest. For instance, this method has been successfully applied for the description of the spectrum of open spin chains, partition functions of elliptic models with domain wall boundaries and scalar product of Bethe vectors. Further applications of this method are also discussed.Functional equations methods are a fundamental part of the theory of Exactly Solvable Models in Statistical Mechanics and they are intimately connected with Baxter's concept of commuting transfer matrices. This concept has culminated in the celebrated Yang-Baxter equation which plays a fundamental role for the construction of quantum integrable systems and also for obtaining their exact solution. Here I shall discuss a proposal that has been put forward in the past years, in which the Yang-Baxter algebra is viewed as a source of functional equations describing quantities of physical interest. For instance, this method has been successfully applied for the description of the spectrum of open spin chains, partition functions of elliptic models with domain wall boundaries and scalar product of Bethe vectors. Further applications of this method are also discussed.

arXiv:1312.6816Journal of Physics Conference SeriesNov 29, 20138 citations

Quantum groups

M. Ruiz-Altaba

These notes correspond rather accurately to the translation of the lectures given at the Fifth Mexican School of Particles and Fields, held in Guanajuato, Gto., in December 1992. They constitute a brief and elementary introduction to quantum symmetries from a physical point of view, along the lines of the forthcoming book by C. Gómez, G. Sierra, and myself: 1) Introduction. 2) Factorizable S‐matrices. 3) Bethe’s diagonalization of spin chain hamiltonians. 4) Integrable vertex models: the six‐vertex model. 5) The Yang‐Baxter algebra. 6) Physical spectrum of the Heisenberg spin chain. 7) Yang‐Baxter algebras and braid groups. 8) Yang‐Baxter algebras and quantum groups. 9) Affine quantum groups. 10) Hopf algebras. 11) The quantum group Uq(G). 12) Comments. 13) References.These notes correspond rather accurately to the translation of the lectures given at the Fifth Mexican School of Particles and Fields, held in Guanajuato, Gto., in December 1992. They constitute a brief and elementary introduction to quantum symmetries from a physical point of view, along the lines of the forthcoming book by C. Gómez, G. Sierra, and myself: 1) Introduction. 2) Factorizable S‐matrices. 3) Bethe’s diagonalization of spin chain hamiltonians. 4) Integrable vertex models: the six‐vertex model. 5) The Yang‐Baxter algebra. 6) Physical spectrum of the Heisenberg spin chain. 7) Yang‐Baxter algebras and braid groups. 8) Yang‐Baxter algebras and quantum groups. 9) Affine quantum groups. 10) Hopf algebras. 11) The quantum group Uq(G). 12) Comments. 13) References.

arXiv:hep-th/9311069AIP conference proceedingsJan 1, 19940 citations

The Six-Vertex Yang-Baxter Groupoid

Daniel Bump, Slava Naprienko

A parametrized Yang-Baxter equation is usually defined to be a map from a group to a set of R-matrices, satisfying the Yang-Baxter commutation relation. These are a mainstay of solvable lattice models. We will show how the parameter space can sometimes be enlarged to a groupoid, and give two examples of such groupoid parametrized Yang-Baxter equations, within the six vertex model. A groupoid parametrized Yang-Baxter equation consists of a groupoid $\mathfrak{G}$ together with a map $π:\mathfrak{G}\to\operatorname{End}(V\otimes V)$ for some vector space $V$ such that the Yang-Baxter commutator $[[ π(u),π(w),π(v)]]=0$ if $u,v\in\mathfrak{G}$ are such that the groupoid composition $w=u\star v$ is defined. An important role is played by an object map $Δ:\mathfrak{G}\to M$ for some set $M$ such that $Δ(u)=Δ(v')$ , $Δ(w)=Δ(v)$ and $Δ(w')=Δ(u')$ , where $v\mapsto v'$ is the groupoid inverse map. There are two main regimes of the six-vertex model: the free-fermionic point, and everything else. For the free-fermionic point, there exists a parametrized Yang-Baxter equation with a large parameter group $\operatorname{GL}(2)\times\operatorname{GL}(1)$ . For non-free-fermionic six-vertex matrices, there are also well-known (group) parametrized Yang-Baxter equations, but these do not account for all possible interactions. Instead we will construct a groupoid parametrized Yang-Baxter equation that accounts for essentially all possible Yang-Baxter equations in the six-vertex model. We will also exhibit a separate groupoid for the five-vertex model. We will show how to construct solvable lattice models based on groupoid parametrized Yang-Baxter equations.A parametrized Yang-Baxter equation is usually defined to be a map from a group to a set of R-matrices, satisfying the Yang-Baxter commutation relation. These are a mainstay of solvable lattice models. We will show how the parameter space can sometimes be enlarged to a groupoid, and give two examples of such groupoid parametrized Yang-Baxter equations, within the six vertex model. A groupoid parametrized Yang-Baxter equation consists of a groupoid $\mathfrak{G}$ together with a map $π:\mathfrak{G}\to\operatorname{End}(V\otimes V)$ for some vector space $V$ such that the Yang-Baxter commutator $[[ π(u),π(w),π(v)]]=0$ if $u,v\in\mathfrak{G}$ are such that the groupoid composition $w=u\star v$ is defined. An important role is played by an object map $Δ:\mathfrak{G}\to M$ for some set $M$ such that $Δ(u)=Δ(v')$, $Δ(w)=Δ(v)$ and $Δ(w')=Δ(u')$, where $v\mapsto v'$ is the groupoid inverse map. There are two main regimes of the six-vertex model: the free-fermionic point, and everything else. For the free-fermionic point, there exists a parametrized Yang-Baxter equation with a large parameter group $\operatorname{GL}(2)\times\operatorname{GL}(1)$. For non-free-fermionic six-vertex matrices, there are also well-known (group) parametrized Yang-Baxter equations, but these do not account for all possible interactions. Instead we will construct a groupoid parametrized Yang-Baxter equation that accounts for essentially all possible Yang-Baxter equations in the six-vertex model. We will also exhibit a separate groupoid for the five-vertex model. We will show how to construct solvable lattice models based on groupoid parametrized Yang-Baxter equations.

arXiv:2503.05960Mar 7, 20250 citations

A general functional definition of the infinite dimensional quantum

R

-matrix satisfying the Yang-Baxter equation is given. A procedure for the extracting a finite dimensional

R

-matrix from the general definition is demonstrated in a particular case when the group

SU(2)

takes place.

\mathfrak{G}

together with a map

π:\mathfrak{G}\to\operatorname{End}(V\otimes V)

for some vector space

V

such that the Yang-Baxter commutator

[[ π(u),π(w),π(v)]]=0

u,v\in\mathfrak{G}

are such that the groupoid composition

w=u\star v

is defined. An important role is played by an object map

Δ:\mathfrak{G}\to M

for some set

M

such that

Δ(u)=Δ(v')

Δ(w)=Δ(v)

and

Δ(w')=Δ(u')

, where

v\mapsto v'

is the groupoid inverse map. There are two main regimes of the six-vertex model: the free-fermionic point, and everything else. For the free-fermionic point, there exists a parametrized Yang-Baxter equation with a large parameter group

\operatorname{GL}(2)\times\operatorname{GL}(1)

. For non-free-fermionic six-vertex matrices, there are also well-known (group) parametrized Yang-Baxter equations, but these do not account for all possible interactions. Instead we will construct a groupoid parametrized Yang-Baxter equation that accounts for essentially all possible Yang-Baxter equations in the six-vertex model. We will also exhibit a separate groupoid for the five-vertex model. We will show how to construct solvable lattice models based on groupoid parametrized Yang-Baxter equations.