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

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.