Finiteness of Nichols Algebras and Nichols (Braided) Lie Algebras

Abstract

It is shown that if $\mathfrak{B}(V)$ is connected Nichols algebra of diagonal type with $\dim V>1$, then $\dim ({\mathfrak{L}}^{-}(V))=\infty$ $($resp. $\dim (\mathfrak{L}(V))=\infty$$)$ $($ resp. $\dim (\mathfrak{B}(V))=\infty$$)$ if and only if $\Delta (\mathfrak{B}(V))$ is an arithmetic root system and the quantum numbers (i.e. the fixed parameters) of generalized Dynkin diagrams of $V$ are of finite order. Sufficient and necessary conditions for $m$-fold adjoint action in $\mathfrak{B}(V)$ equal to zero, viz. ${\overline{l}}_{x_i}^m[x_j{]}^{-}=0$ for $x_i,x_j\in \mathfrak{B}(V)$, are given.

24 pages