Affine super Yangian and Weyl groupoid(extended)

Abstract

We study affine super Yangians of type $\widehat{\mathfrak{sl}}(m|n)$ over $\mathbb{C}[\hslash ]$ for arbitrary cyclic parity sequences, with $m,n\ge 2$ and $m\ne n$. We prove the equivalence of the full current and degree-zero-and-one presentations by reconstructing all current relations. A finite-type RTT calculation and a cut-diagram argument give explicit normalized quantum odd reflections, including the affine vertex. Together with even braid lifts and diagonal sign automorphisms, these maps satisfy the defining relations of an extended reflection groupoid. We compare the positive-root tensors under odd reflection and identify their difference with the reduced coproduct of the quadratic Cartan correction. Starting from the distinguished coproduct, this establishes the prescribed coassociative coproduct for every cyclic parity sequence. We specify the compatible degreewise completions needed for coproduct comparison at the affine vertex.