Nichols algebras over classical Weyl groups, Fomin-Kirillov algebras and Lyndon basis

TL;DR

This paper investigates the structure and finiteness conditions of Nichols algebras over classical Weyl groups, establishing connections with Fomin-Kirillov algebras, and classifying their representations and bases.

Contribution

It provides new classifications of Nichols algebras over classical Weyl groups, relates them to Fomin-Kirillov algebras, and characterizes their finite-dimensional cases and bases.

Findings

Most conjugacy classes in W(B_n) and W(D_n) are of type D.

Except in three cases, Nichols algebras over classical Weyl groups are infinite dimensional.

Established relationships between Fomin-Kirillov algebras and Nichols algebras.

Abstract

We show that except in several cases conjugacy classes of classical Weyl groups $W(B_n)$ and $W(D_n)$ are of type {\rm D}. We prove that except in three cases Nichols algebras of irreducible Yetter-Drinfeld ({\rm YD} in short )modules over the classical Weyl groups are infinite dimensional. We establish the relationship between Fomin-Kirillov algebra $\mathcal E_n$ and Nichols algebra $\mathfrak{B} ({\mathcal O}_{{(1, 2)}} , \epsilon \otimes {\rm sgn})$ of transposition over symmetry group by means of quiver Hopf algebras. We generalize {\rm FK } algebra. The characteristic of finiteness of Nichols algebras in thirteen ways and of {\rm FK } algebras ${\mathcal E}_n$ in nine ways is given. All irreducible representations of finite dimensional Nichols algebras %({\rm FK } algebras ${\mathcal E}_n$) and a complete set of hard super- letters of Nichols algebras of finite Cartan types are found. The sufficient and necessary condition for Nichols algebra $\mathfrak B(M)$ of reducible {\rm YD} module $M$ over $A \rtimes \mathbb{S}_n$ with ${\rm supp } (M) \subseteq A$ to be finite dimensional is given. % Some conditions for a braided vector space to become a {\rm YD} module over finite commutative group are obtained. It is shown that hard braided Lie Lyndon word, standard Lyndon word, Lyndon basis path, hard Lie Lyndon word and standard Lie Lyndon word are the same with respect to $ \mathfrak B(V)$, Cartan matrix $A_c$ and $U(L^+)$, respectively, where $V$ and $L$ correspond to the same finite Cartan matrix $A_c$.