Solutions to the Quantum Yang-Baxter Equation with Extra Non-Additive Parameters

TL;DR

This paper introduces a systematic method for constructing solutions to the Quantum Yang-Baxter Equation that depend on additional non-additive continuous parameters, expanding the class of integrable models.

Contribution

It develops a technique to generate R-matrices with extra non-additive parameters using non-compact and superalgebra representations.

Findings

Constructed R-matrices for $U_q(su(1,1))$, $U_q(gl(1|1))$, and $U_q(gl(2|1))$.

Demonstrated the dependence of solutions on extra continuous parameters.

Extended the framework of solutions to include non-additive parameter dependence.

Abstract

We present a systematic technique to construct solutions to the Yang-Baxter equation which depend not only on a spectral parameter but in addition on further continuous parameters. These extra parameters enter the Yang-Baxter equation in a similar way to the spectral parameter but in a non-additive form. We exploit the fact that quantum non-compact algebras such as $U_q(su(1,1))$ and type-I quantum superalgebras such as $U_q(gl(1|1))$ and $U_q(gl(2|1))$ are known to admit non-trivial one-parameter families of infinite-dimensional and finite dimensional irreps, respectively, even for generic $q$. We develop a technique for constructing the corresponding spectral-dependent R-matrices. As examples we work out the the $R$-matrices for the three quantum algebras mentioned above in certain representations.