---
title: Rational Kashiwara–Miwa Model
url: https://www.emergentmind.com/topics/rational-kashiwara-miwa-model
type: topic
---

# Rational Kashiwara–Miwa Model

The rational Kashiwara–Miwa model is an example of an Ising-type integrable model of the statistical physics, related to the six-vertex trigonometric $R$-matrix. Its two-spin edge weights are expressed in the terms of $q$-products, its spins are arbitrary integers, and $|q|<1$. In "On algebraic structures underlying the rational Kashiwara-Miwa-type models" [2508.12537], the model is developed from the $q$-oscillator algebra, from representations of the $q$-oscillator algebra, and from the co-product of the $q$-oscillator algebra, culminating in explicit two-spin weights and their star–triangle relation.

## 1. Model class and algebraic scope

The exposition is organized as a self-contained development of the rational Kashiwara–Miwa model, beginning with the $q$-oscillator algebra and ending with the explicit two-spin weights and their star–triangle relation. The central claim is that the model’s two-spin edge weights in closed $q$-product form, the six-vertex trigonometric $R$-matrix, the $q$-oscillator algebra, the Hopf coproduct, the $L$-operator, and the star–triangle relation are derived from the underlying $q$-oscillator Hopf algebra and its infinite-dimensional “spin” representations [2508.12537].

This places the model simultaneously in two standard integrability languages. On one side stands the six-vertex trigonometric $R$-matrix and the $RLL$ relation. On the other stands the interaction-round-a-face formulation encoded by the star–triangle identity. The presentation makes these two descriptions equivalent through the usual vertex–IRF correspondence. A plausible implication is that the rational Kashiwara–Miwa model is best understood not as an isolated family of Boltzmann weights, but as a representation-theoretic realization of a specific Hopf-algebraic structure.

## 2. Generalised $q$-oscillator algebra

The basic algebraic object is the “generalised” $q$-oscillator algebra $\mathcal{O}_q[\mathcal{K},\mathcal{K}']$, generated by $\mathcal{E}^+$, $\mathcal{E}^-$, $\mathcal{K}$, and $\mathcal{K}'$ subject to
$$
\begin{aligned}
\mathcal{K}\,\mathcal{E}^{\pm}&=q^{\pm1}\,\mathcal{E}^{\pm}\,\mathcal{K},&
\mathcal{K}'\,\mathcal{E}^{\pm}&=q^{\pm1}\,\mathcal{E}^{\pm}\,\mathcal{K}',&
[\mathcal{K},\mathcal{K}']&=0,\\
\mathcal{E}^{+}\,\mathcal{E}^{-}&=1-q^{-1}\,\mathcal{K}\,\mathcal{K}',&
\mathcal{E}^{-}\,\mathcal{E}^{+}&=1-q\,\mathcal{K}\,\mathcal{K}'.
\end{aligned}
$$

In the special case $\mathcal{K}\mathcal{K}'=q$, one recovers the standard $q$-oscillator relation
$$
q\,\mathcal{E}^+\,\mathcal{E}^-\;-\;q^{-1}\,\mathcal{E}^-\,\mathcal{E}^+
\;=\;q-q^{-1}.
$$
This identifies the generalised algebra as an extension of the standard $q$-oscillator setting rather than a different construction.

The same structure is rewritten in the more familiar variables
$$
a=\mathcal{E}^-,\qquad a^\dagger=\mathcal{E}^+,\qquad q^N=\mathcal{K}\mathcal{K}'\,q^{-1},
$$
for which the familiar relations hold:
$$
q\,a^\dagger a-q^{-1}a\,a^\dagger=q-q^{-1},\qquad
q^N a^\dagger = q\,a^\dagger q^N,\qquad
q^N a=q^{-1}a\,q^N.
$$
This reformulation is significant because it exhibits the algebra in a notation standard for oscillator-type representations while preserving the model-specific role of $\mathcal{K}$ and $\mathcal{K}'$.

## 3. Hopf structure, $L$-operators, and the six-vertex $R$-matrix

The coproduct is given by the algebra homomorphism
$$
\begin{aligned}
\Delta(\mathcal{E}^-)&=\mathcal{E}^-\otimes\mathcal{E}^-\;-\;\mathcal{K}\otimes\mathcal{K}',&
\Delta(\mathcal{K})&=\mathcal{E}^-\otimes\mathcal{K}+\mathcal{K}\otimes\mathcal{E}^+,\\
\Delta(\mathcal{K}')&=\mathcal{K}'\otimes\mathcal{E}^-+\mathcal{E}^+\otimes\mathcal{K}',&
\Delta(\mathcal{E}^+)&=\mathcal{E}^+\otimes\mathcal{E}^+\;-\;\mathcal{K}'\otimes\mathcal{K},
\end{aligned}
$$
and this makes $\mathcal{O}_q[\mathcal{K},\mathcal{K}']$ a Hopf algebra. Equivalently, with
$$
L=
\begin{pmatrix}
\mathcal{E}^- & -\,\mathcal{K}\\
\mathcal{K}' & \mathcal{E}^+
\end{pmatrix},
$$
one has $\Delta(L)=L\dot\otimes L$.

The spectral-parameter-twisted $L$-operator is
$$
L(\lambda)=
\begin{pmatrix}
\mathcal{E}^- & -\,\mathcal{K}/\lambda\\
\lambda\,\mathcal{K}' & \mathcal{E}^+
\end{pmatrix}
=\mathrm{diag}(1,\lambda)\,L\,\mathrm{diag}(1,\lambda)^{-1}.
$$
The trigonometric six-vertex $R$-matrix is defined by
$$
R(\lambda,\mu)=
\begin{pmatrix}
[\,q\,\mu/\lambda\,]&0&0&0\\
0&[\,\mu/\lambda\,]&[\,q\,]&0\\
0&[\,q\,]&[\,\mu/\lambda\,]&0\\
0&0&0&[\,q\,\mu/\lambda\,]
\end{pmatrix},
\qquad [z]=z-z^{-1},
$$
and satisfies the intertwining relation
$$
R(\lambda,\mu)\,\bigl(L(\lambda)\otimes I\bigr)\,\bigl(I\otimes L(\mu)\bigr)
=
\bigl(I\otimes L(\mu)\bigr)\,\bigl(L(\lambda)\otimes I\bigr)\,R(\lambda,\mu).
$$

In the oscillator variables, the coproduct assumes the standard-looking form
$$
\Delta(a)=a\otimes 1+q^N\otimes a,\qquad
\Delta(a^\dagger)=a^\dagger\otimes q^{-N}+1\otimes a^\dagger,\qquad
\Delta(q^N)=q^N\otimes q^N.
$$
This gives the Hopf structure a direct representation-theoretic use in tensor-product constructions.

## 4. Infinite-dimensional spin representation

To allow arbitrary integer “spins,” the construction uses a homomorphism of $\mathcal{O}_q[\mathcal{K},\mathcal{K}']$ into the Weyl algebra generated by unitary $u,v$ with $u\,v=q\,v\,u$. On the basis $\{|m\rangle,\;m\in\mathbb{Z}\}$,
$$
u\,|m\rangle = |m\rangle\,(\gamma q^m),\qquad
v\,|m\rangle = |m+1\rangle.
$$
The representation $\mathbb{V}_\gamma$ is then defined by explicit actions of $\pi^{(\gamma)}(\mathcal{K}_0)$ and $\pi^{(\gamma)}(\mathcal{E}^\pm)$ on $|m\rangle$, where $\mathcal{K}_0$ is the common value of $\omega^{-1}\mathcal{K}=\omega\,\mathcal{K}'$ [2508.12537].

One explicit formula in this representation is
$$
\pi^{(\gamma)}(\mathcal{K}_0)\,|m\rangle
=\frac{|m+1\rangle-|m-1\rangle}{[\gamma q^m]},
$$
with
$$
[\gamma q^m]=\gamma q^m-\gamma^{-1}q^{-m}.
$$
The paper states that $m\in\mathbb{Z}$ is arbitrary and that, for generic $\gamma$, the representation is irreducible. This suggests that the model is formulated here in an infinite-dimensional local-state setting rather than in a finite-spin truncation.

The representation is used to diagonalise
$$
\mathcal{H}(\mu)=\mathcal{K}_0^{-1}\!\bigl(\mu^{-1}\mathcal{E}^--\mu\mathcal{E}^+\bigr),
$$
with eigenvalue equation
$$
\mathcal{H}(\mu)\,|\mu,m\rangle=(\xi+\xi^{-1})\,|\mu,m\rangle,\qquad \xi=\gamma q^m.
$$
This diagonalisation is the immediate source of the two-spin Boltzmann weights.

## 5. Two-spin edge weights and single-spin measure

The similarity operator $\hat V_{\mu,\mu'}:|\mu',m'\rangle\mapsto|\mu,m\rangle$ has matrix elements that become the two-spin Boltzmann weight:
$$
W_x(m,n)=\bigl\langle \mu,m\big|\mu',n\bigr\rangle\Big|_{\mu'/\mu=x}.
$$
A direct solution of the functional recurrences yields the closed form
$$
W_x(m,n)=\bigl(\tfrac{q}{x}\bigr)^{2m}\;
(x;q^2)_{\,m-n}\;(q^2/x;q^2)_{\,m-n}\;
(\gamma^2x;q^2)_{\,m+n}\;(q^2/(\gamma^2x);q^2)_{\,m+n},
$$
valid for all $m,n\in\mathbb{Z}$ [2508.12537].

Here
$$
(u;q^2)_k=\prod_{j=0}^{k-1}(1-u\,q^{2j}),
$$
so the weights are explicitly written in terms of finite $q$-products. This is the concrete realization of the statement that the model’s two-spin edge weights are expressed in the terms of $q$-products.

The associated single-spin measure is
$$
S_m=\frac{[\gamma q^{2m}]}{[\gamma]},
\qquad
[\gamma q^{2m}]=\gamma q^{2m}-\gamma^{-1}q^{-2m}.
$$
The pair $(W_x,S_m)$ is the data entering the star–triangle identity. In this formulation the spin variables are integers, and the dependence on the spectral and representation parameters is carried by $x$, $q$, and $\gamma$.

## 6. Star–triangle relation, inversion, and vertex–IRF correspondence

The weights $W_x(m,n)$ and $S_m$ satisfy, for any $(x,y,z)$ with $x\,y\,z=q$, the star–triangle identity
$$
\sum_{d\in\mathbb{Z}}
S_d\;
W_x(a,d)\;
W_y(b,d)\;
W_z(c,d)
=
\frac{\kappa(x)\,\kappa(y)\,\kappa(z)}{\kappa(1)}
\,W_{q/x}(b,c)\,W_{q/y}(a,c)\,W_{q/z}(a,b),
$$
with a scalar prefactor $\kappa(x)$ expressed in terms of infinite $q$-products. For $\gamma=1$, one example given is
$$
\kappa(x)=\frac{(q^2x^2;q^4)_\infty}{(q^4/x^2;q^4)_\infty}.
$$

The special case $y=1/x$ yields the inversion relation
$$
\sum_d S_d\,W_x(a,d)\,W_{1/x}(d,b)=\Phi(x)\,\delta_{a,b}.
$$
This identity gives a standard consistency reduction of the star–triangle relation.

When the $L$-operator $L(\lambda)$ is represented in the spin-basis $\pi^{(\gamma)}_{\omega,\mu}$, the solution of the $RLL$ relation is equivalent to the star–triangle relation via the usual vertex–IRF correspondence. In particular, the box-decomposition
$$
R=\overline V_{z}\;\bigl(V_{x/y}\otimes V_{y/x}\bigr)\;\overline V_{z'},
\qquad
z,\;z'\;\text{fixed by }x,y,
$$
reproduces the six-vertex $R$-matrix in this infinite-dimensional setting [2508.12537].

Taken together, these constructions show that the rational Kashiwara–Miwa model is encoded by a tightly linked system of algebraic objects: the generalised $q$-oscillator algebra, its Hopf coproduct, the spectral-parameter-dependent $L$-operator, the six-vertex trigonometric $R$-matrix, the infinite-dimensional spin representation, the closed $q$-product Boltzmann weights, and the star–triangle relation. The presentation makes explicit that these are not separate ingredients but parts of a single underlying algebraic framework.

Source: https://www.emergentmind.com/topics/rational-kashiwara-miwa-model