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arXiv · 2310.11855

On Nichols algebras associated to Near-rack solutions of the Yang-Baxter equation

Yuxing Shi
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Abstract

Let $(X, r)$ be any set-theoretical non-degenerate solution of the Yang-Baxter equation and $(X, \tilde r)$ be the derived solution of $(X, r)$. As for any braided vector space $(W_{X, r}, c)$ associated to $(X, r)$, is it possible to find some braided vector space $(W_{X, \tilde r}, \tilde c)$ which is t-equivalent to $(W_{X, r}, c)$? In case that $(X, r)$ is a near-rack solution, we give a sufficient condition to make an affirmative answer to the question. Examples of t-equivalence are constructed, hence finite dimensional Nichols algebras are obtained. In particular, all finite dimensional Nichols algebras associated to involutive near-rack solutions are classified.

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Yuxing Shi. 2023-10-18. On Nichols algebras associated to Near-rack solutions of the Yang-Baxter equation. https://arxiv.org/abs/2310.11855

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