Journal of Lie Theory

no. 2
Structures of Nichols (Braided) Lie Algebras of Diagonal Type
Weicai Wu1, 2; Jing Wang3; Shouchuan Zhang4; Yao-Zhong Zhang5
1Dept. of Mathematics, Zhejiang University, Hangzhou 310007, P. R. China
2School of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, P. R. China
3College of Science, Beijing Forestry University, Beijing 100083, P. R. China
4Dept. of Mathematics, Hunan University, Changsha 410082, P. R. China
5School of Mathematics and Physics, University of Queensland, Brisbane 4072, Australia
Journal of Lie Theory, Volume 28 (2018) no. 2, pp. 357-380
\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 $\dim V=2$.
DOI: 10.5802/jolt.1006
Classification: 16W30, 16G10
Keywords: Braided vector space, Nichols algebra, Nichols braided Lie algebra, graph

Weicai Wu  1 , 2 ; Jing Wang  3 ; Shouchuan Zhang  4 ; Yao-Zhong Zhang  5

1 Dept. of Mathematics, Zhejiang University, Hangzhou 310007, P. R. China
2 School of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, P. R. China
3 College of Science, Beijing Forestry University, Beijing 100083, P. R. China
4 Dept. of Mathematics, Hunan University, Changsha 410082, P. R. China
5 School of Mathematics and Physics, University of Queensland, Brisbane 4072, Australia 
Weicai Wu; Jing Wang; Shouchuan Zhang; Yao-Zhong Zhang. Structures of Nichols (Braided) Lie Algebras of Diagonal Type. Journal of Lie Theory, Volume 28 (2018) no. 2, pp. 357-380. doi: 10.5802/jolt.1006
@article{JOLT_2018_28_2_a3,
     author = {Weicai Wu and Jing Wang and Shouchuan Zhang and Yao-Zhong Zhang},
     title = {Structures of {Nichols} {(Braided)} {Lie} {Algebras} of {Diagonal} {Type}},
     journal = {Journal of Lie Theory},
     pages = {357--380},
     year = {2018},
     volume = {28},
     number = {2},
     doi = {10.5802/jolt.1006},
     zbl = {1391.16038},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1006/}
}
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