On Nichols (braided) Lie algebras

TL;DR

This paper establishes the equivalence of finite-dimensionality between Nichols algebras and Nichols braided Lie algebras, explores their structure in specific cases, and provides dimension estimates and examples of unusual Lie algebras.

Contribution

It proves key equivalences and structural properties of Nichols and Nichols braided Lie algebras, including dimension criteria and specific case analyses.

Findings

Finite-dimensional Nichols algebra iff its Nichols braided Lie algebra is finite-dimensional.

In rank 2, Nichols algebra with an arithmetic root system decomposes as a direct sum involving its Lie algebra.

Conditions under which Nichols algebra has infinite dimension, related to twisting equivalence and Lie algebra dimensions.

Abstract

We prove {\rm (i)} Nichols algebra $\mathfrak B(V)$ of vector space $V$ is finite-dimensional if and only if Nichols braided Lie algebra $\mathfrak L(V)$ is finite-dimensional; {\rm (ii)} If the rank of connected $V$ is $2$ and $\mathfrak B(V)$ is an arithmetic root system, then $\mathfrak B(V) = F \oplus \mathfrak L(V);$ and {\rm (iii)} if $\Delta (\mathfrak B(V))$ is an arithmetic root system and there does not exist any $m$-infinity element with $p_{uu} \not= 1$ for any $u \in D(V)$, then $\dim (\mathfrak B(V) ) = \infty$ if and only if there exists $V'$, which is twisting equivalent to $V$, such that $ \dim (\mathfrak L^ - (V')) = \infty.$ Furthermore we give an estimation of dimensions of Nichols Lie algebras and two examples of Lie algebras which do not have maximal solvable ideals.