Structures of Nichols (braided) Lie algebras of diagonal type

Abstract

Let $V$ be a braided vector space of diagonal type. Let $\mathfrak{B}(V)$, ${\mathfrak{L}}^{-}(V)$ and $\mathfrak{L}(V)$ be the Nichols algebra, Nichols Lie algebra and Nichols braided Lie algebra over $V$, respectively. We show that a monomial belongs to $\mathfrak{L}(V)$ if and only if that this monomial is connected. We obtain the basis for $\mathfrak{L}(V)$ of arithmetic root systems and the dimension for $\mathfrak{L}(V)$ of finite Cartan type. We give the sufficient and necessary conditions for $\mathfrak{B}(V)=F\oplus {\mathfrak{L}}^{-}(V)$ and ${\mathfrak{L}}^{-}(V)=\mathfrak{L}(V)$. We obtain an explicit basis of ${\mathfrak{L}}^{-}(V)$ over quantum linear space $V$ with $\dim V=2$.

23 pages. Version to appear in Journal of Lie Theory