Zamolodchikov–Faddeev algebras for Yangian doubles at level 1

The representation theory of centrally extended Yangian doubles is investigated. The intertwining operators are constructed for infinite dimensional representations of $\widehat{DY(\mathfrak{sl}_2)}$, which are deformed analogs of the highest weight representations of the affine algebra $\widehat{\mathfrak{sl}}_2$ at level 1. We give bosonized expressions for the intertwining operators and verify that they generate an algebra isomorphic to the Zamolodchikov–Faddeev algebra for the $SU(2)$-invariant Thirring model. From them, we compose $L$-operators by Miki's method and verify that they coincide with $L$-operators constructed from the universal $\mathcal R$-matrix. The matrix elements of the product of these operators are calculated explicitly and are shown to satisfy the quantum (deformed) Knizhnik–Zamolodchikov equation associated with the universal $\mathcal R$-matrix for $\widehat{DY(\mathfrak{sl}_2)}$.