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Journal of Lie Theory 28 (2018), No. 2, 357--380
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

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

Yao-Zhong Zhang
School of Mathematics and Physics, University of Queensland, Brisbane 4072, Australia


\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

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