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
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
Yao-Zhong Zhang
School of Mathematics and Physics, University of Queensland, Brisbane 4072, Australia
yzz@maths.uq.edu.au
[Abstract-pdf]
\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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