Classicalrmatrix of thesu(2|2)super Yang-Mills spin chain

Abstract

In this note we straightforwardly derive and make use of the quantum $R$ matrix for the $\mathfrak{s}\mathfrak{u}(2|2)$ super Yang-Mills spin chain in the manifest $\mathfrak{s}\mathfrak{u}(1|2)$-invariant formulation, which solves the standard quantum Yang-Baxter equation, in order to obtain the correspondent (undressed) classical $r$ matrix from the first order expansion in the ``deformation'' parameter $2\ensuremath{\pi}/\sqrt{\ensuremath{\lambda}}$ and check that this last solves the standard classical Yang-Baxter equation. We analyze its bialgebra structure, its dependence on the spectral parameters, and its pole structure. We notice that it still preserves an $\mathfrak{s}\mathfrak{u}(1|2)$ subalgebra, thereby admitting an expression in terms of a combination of projectors, which spans only a subspace of $\mathfrak{s}\mathfrak{u}(1|2)\ensuremath{\bigotimes}\mathfrak{s}\mathfrak{u}(1|2)$. We study the residue at its simple pole at the origin and comment on the applicability of the classical Belavin-Drinfeld type of analysis.