Shifted quantum affine superalgebras

Abstract

For a quantum affine superalgebra with a fixed Drinfeld current presentation, we define shifted versions by modifying the leading degrees of the Cartan--Drinfeld currents while retaining the remaining current and Serre relations. We then study the standard-parity type A algebras $\mathcal U_q^{μ_+,μ_-}(\widehat{\mathfrak{sl}}_{m|n})$. For arbitrary shifts and $m,n\ge1$, $m\ne n$, we prove a triangular decomposition. When $m,n>1$, we obtain PBW--Drinfeld bases for arbitrary shifts and a Levendorskii-type finite presentation of the one-sided algebra $\mathcal U_q^{0,μ}$ for every antidominant $μ$. For antidominant two-sided shifts, we construct shifted RTT realizations and prove the corresponding Ding--Frenkel isomorphisms. We construct a shifted formal Drinfeld coproduct for arbitrary two-sided shifts, coassociative in a completed tensor product. Shifted RTT realizations yield non-completed one-sided coproducts in the antidominant region, made explicit on finitely many generators by the finite presentation; injective shift homomorphisms extend them to arbitrary one-sided shifts. Their ternary coassociativity holds whenever the middle shift is antidominant, including boundary cases. Finally, for two-sided antidominant shifts $μ_+,μ_-\inΛ^-$, a shifted full general-linear Ding--Frenkel isomorphism identifies the special-linear simple-current algebra as the fixed algebra under common scalar rescalings and yields coassociative two-sided non-completed coproducts.