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We establish the relationship among Nichols algebras, Nichols braided Liealgebras and Nichols Lie algebras. We prove two results: (i) Nichols algebra$\mathfrak B(V)$ is finite-dimensional if and only if Nichols braided Liealgebra $\mathfrak L(V)$ is finite-dimensional if there does not exist any$m$-infinity element in $\mathfrak B(V)$; (ii) Nichols Lie algebra $\mathfrakL^-(V)$ is infinite dimensional if $ D^-$ is infinite. We give the sufficientconditions for Nichols braided Lie algebra $\mathfrak L(V)$ to be a homomorphicimage of a braided Lie algebra generated by $V$ with defining relations.

Relationship between Nichols braided Lie algebras and Nichols algebras