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arXiv 2106.00552math.QAmath.CO
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关于Nichols双特征代数

On Nichols bicharacter algebras

Weicai Wu

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16pages
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摘要：

本文定义了两种Lie运算，并由此定义了双特征代数、Nichols双特征代数、量子Nichols双特征代数等，得到了（i）具有$\dim V=2$的量子线性空间$V$;（ii）具有$\dim V=2$和$p_{1,1}=p_{2,2}= -1$的对角型连通辫向量$V$上$\mathfrak L(V)$ {\tiny $_{R}$ }和$\mathfrak L(V)$ {\tiny $_{L}$ }的显式基.分别给出了$\mathfrak L(V)$ { \tiny $_{R}$ } $= \mathfrak L(V)$，$\mathfrak L(V)$ { \tiny $_{L}$ } $= \mathfrak L(V)$，$\mathfrak B(V) = F\oplus \mathfrak L(V)$ { \tiny $_{R}$ }和$\mathfrak B(V) = F\oplus \mathfrak L(V)$ { \tiny $_{L}$ }的充分必要条件.证明了若$\mathfrak B(V)$是具有$\dim V>1$的对角型连通Nichols代数，则$\mathfrak B(V)$是有限维的当且仅当$\mathfrak L(V)$ { \tiny $_{L}$ }是有限维的当且仅当$\mathfrak L(V)$ { \tiny $_{R}$ }是有限维的.

英文摘要：

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.

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