Finiteness of Nichols Algebras and Nichols (Braided) Lie Algebras

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 $x_i,~x_jin mathfrak B(V)$, are given.