The supersymmetric affine Yangian

Abstract The affine Yangian of gl1 is known to be isomorphic to W1+∞ $$ {\mathcal{W}}_{1+\infty } $$, the W-algebra that characterizes the bosonic higher spin — CFT duality. In this paper we propose some of the defining relations of the Yangian that are relevant for the N=2 $$ \mathcal{N}=2 $$ superconformal version of W1+∞ $$ {\mathcal{W}}_{1+\infty } $$. Our construction is based on the observation that the N=2 $$ \mathcal{N}=2 $$ superconformal W1+∞ $$ {\mathcal{W}}_{1+\infty } $$ algebra contains two commuting bosonic W1+∞ algebras, and that the additional generators transform in bi-minimal representations with respect to these two algebras. The corresponding affine Yangian can therefore be built up from two affine Yangians of gl1 $$ \mathfrak{g}{\mathfrak{l}}_1 $$ by adding in generators that transform appropriately