On Nichols (braided) Lie algebras

摘要

We prove (i) Nichols algebra B(V) of vector space V is finite dimensional if and only if Nichols braided Lie algebra L(V) is finite dimensional; (ii) if the rank of connected V is 2 and B(V) is an arithmetic root system, then B(V) = F circle plus L(V); and (iii) if Delta(B(V)) is an arithmetic root system and there does not exist any m-infinity element with p(uu) not equal 1 for any u is an element of D(V), then dim (B(V)) = infinity if and only if there exists V', which is twisting equivalent to V, such that dim(L-(V')) = infinity. Furthermore, we give an estimation of dimensions of Nichols Lie algebras and two examples of Lie algebras which do not have maximal solvable ideals.