On Nichols (braided) Lie algebras

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 $p_{uu}\ne 1$ for any $u\in D(V)$, then $\dim (\mathfrak{B}(V))=\infty$ if and only if there exists $V^{\prime }$, which is twisting equivalent to $V$, such that $\dim ({\mathfrak{L}}^{-}(V^{\prime }))=\infty .$ Furthermore we give an estimation of dimensions of Nichols Lie algebras and two examples of Lie algebras which do not have maximal solvable ideals.

29 Pages; Substantially revised version; To appear in International Journal of Mathematics

References in corpus (1)

• Nichols algebras of unidentified diagonal type

Cited by in corpus (3)

• Finiteness of Nichols Algebras and Nichols (Braided) Lie Algebras
• On Nichols bicharacter algebras
• Basis of Nichols Braided Lie Algebras