Relationship between Nichols Braided Lie Algebras and Nichols algebras

摘要

We establish the relationship among Nichols algebras, Nichols braided Lie algebras and Nichols Lie algebras. We prove two results: (i) The Nichols algebra B(V) is finite-dimensional if and only if the Nichols braided Lie algebra L(V) is finite-dimensional if there does not exist any m-infinity element in B(V); (ii) the Nichols Lie algebra L-(V) is infinite dimensional if D- is infinite. We give sufficient conditions for the Nichols braided Lie algebra L(V) to be a homomorphic image of a braided Lie algebra generated by V with defining relations.