Shifted quantum affine superalgebras

For a quantum affine superalgebra with a fixed Drinfeld current presentation, we study shifted quantum affine superalgebras through a two-sided shifted Drinfeld current presentation, in which the leading degrees of the Cartan–Drinfeld currents are modified while the remaining current and Serre relations are retained. We then focus on the standard-parity type A algebras ${\mathcal{U}}_q^{{\mu }_{+},{\mu }_{-}}({\widehat{\mathfrak{s}\mathfrak{l}}}_{m\mathrm{\mid }n})$. For arbitrary shifts and $m,n\ge 1$, $m\mathrm{\ne }n$, 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 ${\mathcal{U}}_q^{0,\mu }$ for every antidominant $\mu$. 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 ${\mu }_{+},{\mu }_{-}\in {\mathrm{\Lambda }}^{-}$, 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.