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

Yang-Baxter algebras and quantum groups

Hector J. de Vega

Nuclear Physics B - Proceedings SupplementsAug 1, 19900 citations

Finite Groups and Quantum Yang-Baxter Equation

Varga, Peter

We construct integrable modifications of 2d lattice gauge theories with finite gauge groups.

arXiv:hep-th/9702157Feb 21, 19970 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.