Journal of Lie Theory 25 (2015), No. 1, 045--063
Copyright Heldermann Verlag 2015
Relationship between Nichols Braided Lie Algebras and Nichols algebras
Weicai Wu
Department of Mathematics, Hunan University, Changsha 410082, P. R. China
weicaiwu@hnu.edu.cn
Shouchuan Zhang
Department of Mathematics, Hunan University, Changsha 410082, P. R. China
sczhang@hnu.edu.cn
Yao-Zhong Zhang
School of Mathematics and Physics, The University of Queensland, Brisbane 4072, Australia
yzz@maths.uq.edu.au
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.
Keywords: Nichols Lie algebra, Nichols algebra, Nichols braided Lie algebra.
MSC: 16W30, 16G10
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