Right Coideal Subalgebras, Hopf Galois Objects, and imath-Hopf Algebras
Abstract
We study right comodule algebras and their connections with Hopf bi-Galois objects and right coideal subalgebras. For a Hopf $2$-cocycle σ on K and a unital right-colinear algebra map Γ:A→ K, we show that Γ:Aσ→ Kσ is an algebra map precisely when σ is trivial on imΓ⊗imΓ. We describe comodule-algebra morphisms Aσ→ Kσ by characters and establish injectivity criteria using the coradical and extensions of characters to convolution-invertible functionals. For coactions on Hopf quotients projecting to the coproduct, character-evaluation maps are injective, and equality of their images is determined by a unique algebra character. For the imath-Hopf algebras Himathτ of Chen--Lu--Pan--Ruan--Wang, with H having a bijective antipode and a Hopf pairing, we obtain a Hopf bi-Galois realization Himathτ↪ Fσ(H⊗ Hcop). We identify algebra characters with nonzero τ-twisted compatible maps χ and prove that the corresponding morphisms ξτ,χ of Chen--Lu--Pan--Ruan--Wang are injective, yielding right coideal realizations in (H⊗ H)imath. We classify the Himathτ up to comodule-algebra isomorphism and determine when these realizations have the same image.
Related
- One-dimensional partial actions of 8-dimensional Hopf algebras and right coideal subalgebras
- Schur--Weyl duality for coideal subalgebras of quantum affine superalgebras of type A
- Classification of twisted Yangians as coideal subalgebras of Yangians
- A Galois connection between subalgebras and tensor subcategories
- Borel subalgebras of Lie algebras of vector fields
BibTeX
@misc{xiong2026right,
title = {{Right Coideal Subalgebras, Hopf Galois Objects, and $\imath$-Hopf Algebras}},
author = {Rongchuan Xiong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19503},
eprint = {2610.19503},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19503},
}
RIS
TY - UNPB
TI - Right Coideal Subalgebras, Hopf Galois Objects, and imath-Hopf Algebras
AU - Rongchuan Xiong
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19503
UR - https://arxiv.org/abs/2610.19503
AB - We study right comodule algebras and their connections with Hopf bi-Galois objects and right coideal subalgebras. For a Hopf $2$-cocycle σ on K and a unital right-colinear algebra map Γ:A→ K, we show that Γ:A_σ→ K^σ is an algebra map precisely when σ is trivial on imΓ⊗imΓ. We describe comodule-algebra morphisms A_σ→ K^σ by characters and establish injectivity criteria using the coradical and extensions of characters to convolution-invertible functionals. For coactions on Hopf quotients projecting to the coproduct, character-evaluation maps are injective, and equality of their images is determined by a unique algebra character. For the imath-Hopf algebras H^imath_τ of Chen--Lu--Pan--Ruan--Wang, with H having a bijective antipode and a Hopf pairing, we obtain a Hopf bi-Galois realization H^imath_τ↪ F_σ(H⊗ H^cop). We identify algebra characters with nonzero τ-twisted compatible maps χ and prove that the corresponding morphisms ξ_τ,χ of Chen--Lu--Pan--Ruan--Wang are injective, yielding right coideal realizations in (H⊗ H)^imath. We classify the H^imath_τ up to comodule-algebra isomorphism and determine when these realizations have the same image.
ER -