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ON NICHOLS BICHARACTER ALGEBRAS
Keywords:
bicharacter algebras, Nichols algebras, Nichols bicharacter algebraDOI:
https://doi.org/10.17654/0972555526005Abstract
In this paper, we define two Lie operations, and with that we define the bicharacter algebras, Nichols bicharacter algebras, etc. We obtain explicit bases for $\mathfrak{L}(V)_R$ and $\mathfrak{L}(V)_L$ over a connected braided vector $V$ of diagonal type with $\operatorname{dim} V=2$ and $p_{1,1}=p_{2,2}=-1$. We give the sufficient and necessary conditions for $\mathfrak{L}(V)_R=\mathfrak{L}(V), \mathfrak{L}(V)_L= \mathcal{L}(V), \quad \mathfrak{B}(V)=F \oplus \mathcal{L}(V)_R$ and $\mathfrak{B}(V)=F \oplus \mathcal{L}(V)_L$, respectively. We show that if $\mathfrak{B}(V)$ is a connected Nichols algebra of diagonal type with $\operatorname{dim} V>1$, then $\mathfrak{B}(V)$ is finite-dimensional if and only if $\mathfrak{L}(V)_L$ is finite-dimensional if and only if $\mathfrak{L}(V)_R$ is finite-dimensional.
Received: April 4, 2025
Accepted: May 30, 2025
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