Infinite Families of Gauge-Equivalent $R$-Matrices and Gradations of Quantized Affine Algebras

TL;DR

The paper explores how different gradations of quantum affine algebras lead to infinitely many gauge-equivalent R-matrices with varying spectral dependences, impacting physical models like Toda field theories.

Contribution

It provides explicit formulas for gauge-equivalent R-matrices derived from gradations of quantum affine algebras U_q(A_1^{(1)}) and U_q(A_2^{(1)}), highlighting their physical significance.

Findings

Infinite gauge-equivalent R-matrices with different spectral dependences.

Explicit formulas for R-matrices in specific quantum affine algebras.

Gradation choice affects physical models such as Toda field theories.

Abstract

Associated with the fundamental representation of a quantum algebra such as $U_q(A_1)$ or $U_q(A_2)$, there exist infinitely many gauge-equivalent $R$-matrices with different spectral-parameter dependences. It is shown how these can be obtained by examining the infinitely many possible gradations of the corresponding quantum affine algebras, such as $U_q(A_1^{(1)})$ and $U_q(A_2^{(1)})$, and explicit formulae are obtained for those two cases. Spectral-dependent similarity (gauge) transformations relate the $R$-matrices in different gradations. Nevertheless, the choice of gradation can be physically significant, as is illustrated in the case of quantum affine Toda field theories.