
Journal of Lie Theory 28 (2018), No. 2, 357380 Copyright Heldermann Verlag 2018 Structures of Nichols (Braided) Lie Algebras of Diagonal Type Weicai Wu Dept. of Mathematics, Zhejiang University, Hangzhou 310007, P. R. China and: School of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, P. R. China weicaiwu@hnu.edu.cn Jing Wang College of Science, Beijing Forestry University, Beijing 100083, P. R. China Shouchuan Zhang Dept. of Mathematics, Hunan University, Changsha 410082, P. R. China sczhang@hnu.edu.cn YaoZhong Zhang School of Mathematics and Physics, University of Queensland, Brisbane 4072, Australia yzz@maths.uq.edu.au [Abstractpdf] \def\B{{\frak B}} \def\L{{\frak L}} Let $V$ be a braided vector space of diagonal type. Let $\B(V)$, $\L^(V)$ and $\L(V)$ be the Nichols algebra, Nichols Lie algebra and Nichols braided Lie algebra over $V$, respectively. We show that a monomial belongs to $\L(V)$ if and only if this monomial is connected. We obtain the basis for $\L(V)$ of arithmetic root systems and the dimension of $\L(V)$ of finite Cartan type. We give the sufficient and necessary conditions for $\B(V) = F\oplus \L^(V)$ and $\L^(V)= \L(V)$. We obtain an explicit basis for $\L^  (V)$ over the quantum linear space $V$ with $\dim V=2$. Keywords: Braided vector space, Nichols algebra, Nichols braided Lie algebra, graph. MSC: 16W30, 16G10 [ Fulltextpdf (355 KB)] for subscribers only. 