On Nichols bicharacter algebras

Abstract: In this paper we define two Lie operations, and with that we define the bicharacter algebras, Nichols bicharacter algebras, quantum Nichols bicharacter algebras, etc. We obtain explicit bases for $\mathfrak L(V)${\tiny ${R}$} and $\mathfrak L(V)${\tiny ${L}$} over (i) the quantum linear space $V$ with $\dim V=2$; (ii) a connected braided vector $V$ of diagonal type with $\dim V=2$ and $p_{1,1}=p_{2,2}= -1$. We give the sufficient and necessary conditions for $\mathfrak L(V)${\tiny ${R}$}$= \mathfrak L(V)$, $\mathfrak L(V)${\tiny ${L}$}$= \mathfrak L(V)$, $\mathfrak B(V) = F\oplus \mathfrak L(V)${\tiny ${R}$} and $\mathfrak B(V) = F\oplus \mathfrak L(V)${\tiny ${L}$}, respectively. We show that if $\mathfrak B(V)$ is a connected Nichols algebra of diagonal type with $\dim V>1$, then $\mathfrak B(V)$ is finite-dimensional if and only if $\mathfrak L(V)${\tiny ${L}$} is finite-dimensional if and only if $\mathfrak L(V)${\tiny ${R}$} is finite-dimensional.