On Type-I Quantum Affine Superalgebras

TL;DR

This paper investigates quantum deformations of type-I affine superalgebras, identifies additional relations needed for their proper definition, and derives new spectral-dependent R-matrices, including solutions generalizing the free-fermion model.

Contribution

It introduces a general method for deriving spectral parameter R-matrices from quantum affine superalgebras and provides explicit R-matrices for $U_q(sl(m|n)^{(1)})$ in various representations.

Findings

Derived new solutions to the spectral Yang-Baxter equation.

Identified additional relations ('extra q-Serre relations') for proper algebra definition.

Constructed R-matrices with two spectral-like parameters, extending free-fermion models.

Abstract

The type-I simple Lie-superalgebras are $sl(m|n)$ and $osp(2|2n)$. We study the quantum deformations of their untwisted affine extensions $U_q(sl(m|n)^{(1)})$ and $U_q(osp(2|2n)^{(1)})$. We identify additional relations between the simple generators (``extra $q$-Serre relations") which need to be imposed to properly define $\uqgh$ and $U_q(osp(2|2n)^{(1)})$. We present a general technique for deriving the spectral parameter dependent R-matrices from quantum affine superalgebras. We determine the R-matrices for the type-I affine superalgebra $U_q(sl(m|n)^{(1)})$ in various representations, thereby deriving new solutions of the spectral-dependent Yang-Baxter equation. In particular, because this algebra possesses one-parameter families of finite-dimensional irreps, we are able to construct R-matrices depending on two additional spectral-like parameters, providing generalizations of the free-fermion model.