Relationship between Nichols braided Lie algebras and Nichols algebras

TL;DR

This paper explores the connections between Nichols algebras and various types of Nichols Lie algebras, establishing criteria for their finite or infinite dimensionality and conditions for their algebraic structure.

Contribution

It provides new criteria linking the finite-dimensionality of Nichols algebras and Nichols braided Lie algebras, and conditions for their algebraic homomorphic images.

Findings

Nichols algebra is finite-dimensional iff Nichols braided Lie algebra is finite-dimensional without m-infinity elements.

Nichols Lie algebra is infinite-dimensional if D^- is infinite.

Conditions for Nichols braided Lie algebra to be a homomorphic image of a generated braided Lie algebra.

Abstract

We establish the relationship among Nichols algebras, Nichols braided Lie algebras and Nichols Lie algebras. We prove two results: (i) Nichols algebra $\mathfrak B(V)$ is finite-dimensional if and only if Nichols braided Lie algebra $\mathfrak L(V)$ is finite-dimensional if there does not exist any $m$-infinity element in $\mathfrak B(V)$; (ii) Nichols Lie algebra $\mathfrak L^-(V)$ is infinite dimensional if $ D^-$ is infinite. We give the sufficient conditions for Nichols braided Lie algebra $\mathfrak L(V)$ to be a homomorphic image of a braided Lie algebra generated by $V$ with defining relations.