Yang-Baxter permutation group actions on distributive Yang-Baxter algebras

Abstract

Let (X,r) be a finite square-free distributive set-theoretic solution of the Yang-Baxter equation, and let 𝒜(X,r) be its associated Yang-Baxter algebra. We prove that 𝒜(X,r) is isomorphic to a skew polynomial algebra and explicitly compute its Nakayama automorphism. We study the natural faithful action of the permutation group 𝒢(X,r) on 𝒜(X,r) by graded algebra automorphisms. We characterize when 𝒢(X,r) is a reflection group and, in that case, describe the invariant subalgebra 𝒜(X,r)^𝒢(X,r) together with its Jacobian, reflection arrangement, and discriminant. We further establish a necessary and sufficient condition for the noncommutative Auslander theorem to hold for the permutation group action on a distributive Yang-Baxter algebra. Finally, for a class of non-trivial distributive Yang-Baxter algebras, we obtain a lower bound for the pertinency of the permutation group action.

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BibTeX RIS

BibTeX

@misc{he2026yangbaxter,
  title = {{Yang-Baxter permutation group actions on distributive Yang-Baxter algebras}},
  author = {Ji-Wei He and Xiaolan Yu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.14220},
  eprint = {2609.14220},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14220},
}

RIS

TY  - UNPB
TI  - Yang-Baxter permutation group actions on distributive Yang-Baxter algebras
AU  - Ji-Wei He
AU  - Xiaolan Yu
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.14220
UR  - https://arxiv.org/abs/2609.14220
AB  - Let (X,r) be a finite square-free distributive set-theoretic solution of the Yang-Baxter equation, and let 𝒜(X,r) be its associated Yang-Baxter algebra. We prove that 𝒜(X,r) is isomorphic to a skew polynomial algebra and explicitly compute its Nakayama automorphism. We study the natural faithful action of the permutation group 𝒢(X,r) on 𝒜(X,r) by graded algebra automorphisms. We characterize when 𝒢(X,r) is a reflection group and, in that case, describe the invariant subalgebra 𝒜(X,r)^𝒢(X,r) together with its Jacobian, reflection arrangement, and discriminant. We further establish a necessary and sufficient condition for the noncommutative Auslander theorem to hold for the permutation group action on a distributive Yang-Baxter algebra. Finally, for a class of non-trivial distributive Yang-Baxter algebras, we obtain a lower bound for the pertinency of the permutation group action.
ER  -

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