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 \( \mathfrak{g}\mathfrak{l}\left( {1\left| 1 \right.} \right) \).