Shifted quantum affine superalgebras

Abstract

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 ${\mathcal{U}}_q^{{\mu }_{+},{\mu }_{-}}({\widehat{\mathfrak{sl}}}_{m|n})$. For arbitrary shifts and $m,n\ge 1$, $m\ne n$, 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 ${\mathcal{U}}_q^{0,\mu }$ 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 ${\mu }_{+},{\mu }_{-}\in {\Lambda }^{-}$, 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.

arXiv comment67 pages