The Yangian of $\mathfrak{s}\mathfrak{l}\left( {n\left| m \right.} \right)$ and its quantum R-matrices

Abstract

In this paper we study Yangians of $\mathfrak{s}\mathfrak{l}\left( {n\left| m \right.} \right)$ superalgebras. We derive the universal R-matrix and evaluate it on the fundamental representation obtaining the standard Yang R-matrix with unitary dressing factors. For m = 0, we directly recover up to a CDD factor the well-known S-matrices for relativistic integrable models with $\mathfrak{s}\mathfrak{u}(n)$ symmetry. Hence, the universal R-matrix found provides an abstract plug-in formula, which leads to results obeying fundamental physical constraints: crossing symmetry, unitrarity and the Yang-Baxter equation. This implies that the Yangian double unifies all desired symmetries into one algebraic structure. In particular, our analysis is valid in the case of $\mathfrak{s}\mathfrak{l}\left( {n\left| n \right.} \right)$, where one has to extend the algebra by an additional generator leading to the algebra $\mathfrak{g}\mathfrak{l}\left( {n\left| n \right.} \right)$. We find two-parameter families of scalar factors in this case and provide a detailed study for $\mathfrak{g}\mathfrak{l}\left( {1\left| 1 \right.} \right)$.