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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.
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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.
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.
Assume that $V$ is a braided vector space with diagonal type. It is shown that a monomial belongs to Nichols braided Lie algebra $mathfrak L(V)$ if and only if this monomial is connected. A basis of Nichols braided Lie algebra and dimension of Nichols braided Lie algebra of finite Cartan type are obtained.
Let $V$ be a braided vector space of diagonal type. Let $mathfrak B(V)$, $mathfrak L^-(V)$ and $mathfrak L(V)$ be the Nichols algebra, Nichols Lie algebra and Nichols braided Lie algebra over $V$, respectively. We show that a monomial belongs to $mathfrak L(V)$ if and only if that this monomial is connected. We obtain the basis for $mathfrak L(V)$ of arithmetic root systems and the dimension for $mathfrak L(V)$ of finite Cartan type. We give the sufficient and necessary conditions for $mathfrak B(V) = Foplus mathfrak L^-(V)$ and $mathfrak L^-(V)= mathfrak L(V)$. We obtain an explicit basis of $mathfrak L^ - (V)$ over quantum linear space $V$ with $dim V=2$.
In this paper we give the relationship between the connected components of pure generalized Dynkin graphs and Nichols braided Lie algebras.
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