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

TL;DR

This paper develops shuffle algebra realizations for super Yangians and quantum affine superalgebras of type A across all Cartan data, providing new algebraic structures and isomorphisms in the superalgebra context.

Contribution

It introduces new Drinfeld realizations of super Yangians for all Dynkin diagrams and extends shuffle algebra realizations to quantum loop superalgebras with arbitrary Dynkin diagrams.

Findings

Super Yangians are isomorphic to Nazarov's super Yangians via RTT realization.

Positive halves of these super Yangians are not pairwise isomorphic.

Shuffle algebra realizations are adapted to the trigonometric case for quantum loop superalgebras.

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.