# Basis of Nichols Braided Lie Algebras

## TL;DR

This paper characterizes monomials in Nichols braided Lie algebras of diagonal type, establishing a connection between monomial connectivity and algebra membership, and provides bases and dimensions for finite Cartan types.

## Contribution

It introduces a criterion for monomials in Nichols braided Lie algebras and computes bases and dimensions for finite Cartan type cases.

## Key findings

- A monomial belongs to Nichols braided Lie algebra if and only if it is connected.
- A basis for Nichols braided Lie algebra is constructed.
- Dimensions of Nichols braided Lie algebra for finite Cartan types are determined.

## Abstract

Assume that $V$ is a braided vector space with diagonal type. It is shown that a monomial belongs to Nichols braided Lie algebra $\mathfrak L(V)$ if and only if this monomial is connected. A basis of Nichols braided Lie algebra and dimension of Nichols braided Lie algebra of finite Cartan type are obtained.