Structures of Nichols (braided) Lie algebras of diagonal type

TL;DR

This paper characterizes the structure of Nichols Lie algebras of diagonal type, providing bases, dimensions, and conditions for their relations, especially for finite Cartan types and quantum linear spaces.

Contribution

It introduces a criterion for monomials in Nichols braided Lie algebras, constructs explicit bases, and determines conditions for algebra decompositions, advancing understanding of Nichols algebra structures.

Findings

Monomials in al L(V) are connected.

Explicit bases for al L(V) in arithmetic root systems.

Dimension formulas for al L(V) of finite Cartan type.

Abstract

Let $V$ be a braided vector space of diagonal type. Let $\mathfrak B(V)$, $\mathfrak L^-(V)$ and $\mathfrak L(V)$ be the Nichols algebra, Nichols Lie algebra and Nichols braided Lie algebra over $V$, respectively. We show that a monomial belongs to $\mathfrak L(V)$ if and only if that this monomial is connected. We obtain the basis for $\mathfrak L(V)$ of arithmetic root systems and the dimension for $\mathfrak L(V)$ of finite Cartan type. We give the sufficient and necessary conditions for $\mathfrak B(V) = F\oplus \mathfrak L^-(V)$ and $\mathfrak L^-(V)= \mathfrak L(V)$. We obtain an explicit basis of $\mathfrak L^ - (V)$ over quantum linear space $V$ with $\dim V=2$.