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 𝒰qμ+,μ-(sl̂m|n). For arbitrary shifts and m,n≥1, m≠ 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 𝒰q0,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-∈Λ-, 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.
Related
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BibTeX
@misc{hu2026shifted,
title = {{Shifted quantum affine superalgebras}},
author = {Piao-Juan Hu and Jian-Rong Li and Yan-Feng Luo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05228},
eprint = {2610.05228},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05228},
}
RIS
TY - UNPB
TI - Shifted quantum affine superalgebras
AU - Piao-Juan Hu
AU - Jian-Rong Li
AU - Yan-Feng Luo
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.05228
UR - https://arxiv.org/abs/2610.05228
AB - 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 𝒰_q^μ_+,μ_-(sl̂_m|n). For arbitrary shifts and m,n≥1, m≠ 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 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_-∈Λ^-, 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.
ER -