Quadratic algebras associated to permutation idempotent solutions of the YBE

TL;DR

This paper investigates quadratic algebras linked to permutation idempotent solutions of the Yang-Baxter equation, revealing their structure, isomorphisms, and properties through algebraic and geometric methods.

Contribution

It introduces canonical algebras for permutation idempotent solutions and analyzes their properties using noncommutative geometry and PBW algebra theory.

Findings

All algebras for a fixed permutation idempotent type are isomorphic.

Characterization of these algebras via Veronese subalgebras and Segre products.

New results on PBW algebras applied to the permutation idempotent case.

Abstract

We study the quadratic algebras $A(K,X,r)$ associated to a class of strictly braided but idempotent set-theoretic solutions $(X,r)$ of the Yang-Baxter or braid relations. In the invertible case, these algebras would be analogues of braided-symmetric algebras or `quantum affine spaces' but due to $r$ being idempotent they have very different properties. We show that all $A(K,X,r)$ for $r$ of a certain permutation idempotent type are isomorphic for a given $n=|X|$, leading to canonical algebras $A(K,n)$. We study the properties of these both via Veronese subalgebras and Segre products and in terms of noncommutative differential geometry. We also obtain new results on general PBW algebras which we apply in the permutation idempotent case.