Shuffle algebra realizations of type $A$ super Yangians and quantum affine superalgebras for all Cartan data

Abstract

We introduce super Yangians of $\mathfrak{gl}(V),\mathfrak{sl}(V)$ (in the new Drinfeld realization) associated to all Dynkin diagrams of $\mathfrak{gl}(V)$, where $V$ is a finite-dimensional super vector space. We show that all of them are isomorphic to the super Yangians introduced by Maxim Nazarov, by identifying them with the corresponding RTT super Yangians. However, their "positive halves" are not pairwise isomorphic, and we obtain the shuffle algebra realizations of those. We adapt the latter to the trigonometric setup by obtaining the shuffle algebra realizations of the "positive halves" of type $A$ quantum loop superalgebras associated to arbitrary Dynkin diagrams.

v1: 18 pages, comments are welcome! This note is a companion to arXiv:1808.09536. v2: 20 pages, minor corrections, some details added. v3: 21 pages, minor corrections, some details and references added