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 $p_{uu} ot= 1$ 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.
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.
The quiver Hopf algebras are classified by means of ramification systems with irreducible representations. This leads to the classification of Nichols algebras over group algebras and pointed Hopf algebras of type one.
