A Galois connection between subalgebras and tensor subcategories

Abstract

Let ℬ be a braided finite tensor category and let A be a simple commutative algebra in ℬ. We construct an order-reversing Galois connection between subalgebras of A and tensor subcategories of ℬ_A. Let ℬ' denote the Müger center and set A':=A∩ℬ'. The closure operators are B↦⟨ B,A'⟩_alg and ℰ↦⟨ℰ,ℬ_A^loc⟩_⊗. Thus the closed subalgebras are those containing A', while the closed tensor subcategories are those containing ℬ_A^loc; equivalently, the fixed-point intervals [A',A]_alg and [ℬ_A^loc,ℬ_A]_⊗ are anti-isomorphic as lattices. For a finite tensor category 𝒞, the canonical algebra in 𝒵(𝒞) yields an anti-isomorphism between its subalgebras and tensor subcategories of 𝒞. When ℬ is nondegenerate, Frobenius extensions correspond to unimodular tensor subcategories. For a Hopf algebra in ℬ, the correspondence specializes to an order-preserving bijection between Hopf ideals and normal left coideal subalgebras.

Right Coideal Subalgebras, Hopf Galois Objects, and imath-Hopf Algebras
Rongchuan Xiong
arXiv14 Oct 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
Invariant von Neumann subalgebras of crossed products and tensor products
Tattwamasi Amrutam, Artem Dudko, Yongle Jiang +1
arXiv13 Oct 2026PreprintCC BY 4.0Full textAnalysis & PDEs Dynamical systemsarXiv
Generalized Rank Weight and Extended Generalized Poset Weight Defined For Codes Over Rings: A Galois Connection Approach
Jianhua Zheng, Yang Xu, Yidan Chen +2
arXiv8 Oct 2026PreprintCC BY 4.0Full textGeneral mathematicsarXiv
Phantom subcategories on blow-ups of Hirzebruch surfaces
Zeyuan He, Yu-Zhe Liu, Mingzhi Sheng +1
arXiv13 Sep 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
Homology and higher-order intermediate categories for proper abelian subcategories
Zheng-Gang Wu, Xianhui Fu
arXiv1 Oct 2026PreprintCC BY-NC-SA 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeX RIS

BibTeX

@misc{shimizu2026galois,
  title = {{A Galois connection between subalgebras and tensor subcategories}},
  author = {Kenichi Shimizu and Harshit Yadav},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04073},
  eprint = {2609.04073},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04073},
  note = {CC BY-SA 4.0},
}

RIS

TY  - UNPB
TI  - A Galois connection between subalgebras and tensor subcategories
AU  - Kenichi Shimizu
AU  - Harshit Yadav
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.04073
UR  - https://arxiv.org/abs/2609.04073
AB  - Let ℬ be a braided finite tensor category and let A be a simple commutative algebra in ℬ. We construct an order-reversing Galois connection between subalgebras of A and tensor subcategories of ℬ_A. Let ℬ' denote the Müger center and set A':=A∩ℬ'. The closure operators are B↦⟨ B,A'⟩_alg and ℰ↦⟨ℰ,ℬ_A^loc⟩_⊗. Thus the closed subalgebras are those containing A', while the closed tensor subcategories are those containing ℬ_A^loc; equivalently, the fixed-point intervals [A',A]_alg and [ℬ_A^loc,ℬ_A]_⊗ are anti-isomorphic as lattices. For a finite tensor category 𝒞, the canonical algebra in 𝒵(𝒞) yields an anti-isomorphism between its subalgebras and tensor subcategories of 𝒞. When ℬ is nondegenerate, Frobenius extensions correspond to unimodular tensor subcategories. For a Hopf algebra in ℬ, the correspondence specializes to an order-preserving bijection between Hopf ideals and normal left coideal subalgebras.
ER  -

Abstract

Related

Cite