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, Tome 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 $\dim V=2$.
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, Tome 28 (2018) no. 2, pp. 357-380. http://geodesic.mathdoc.fr/item/JOLT_2018_28_2_a3/
@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},
url = {http://geodesic.mathdoc.fr/item/JOLT_2018_28_2_a3/}
}
TY - JOUR
AU - Weicai Wu
AU - Jing Wang
AU - Shouchuan Zhang
AU - Yao-Zhong Zhang
TI - Structures of Nichols (Braided) Lie Algebras of Diagonal Type
JO - Journal of Lie Theory
PY - 2018
SP - 357
EP - 380
VL - 28
IS - 2
UR - http://geodesic.mathdoc.fr/item/JOLT_2018_28_2_a3/
ID - JOLT_2018_28_2_a3
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%A Shouchuan Zhang
%A Yao-Zhong Zhang
%T Structures of Nichols (Braided) Lie Algebras of Diagonal Type
%J Journal of Lie Theory
%D 2018
%P 357-380
%V 28
%N 2
%U http://geodesic.mathdoc.fr/item/JOLT_2018_28_2_a3/
%F JOLT_2018_28_2_a3
