On Nichols bicharacter algebras

TL;DR

This paper introduces new Lie operations to define Nichols bicharacter algebras, provides explicit bases for specific cases, and characterizes conditions for their equality and finite-dimensionality, advancing the understanding of Nichols algebra structures.

Contribution

It defines Nichols bicharacter algebras using new Lie operations and characterizes their bases and finite-dimensionality conditions in specific quantum and braided vector space cases.

Findings

Explicit bases for $rak L(V)_R$ and $rak L(V)_L$ in certain cases.

Necessary and sufficient conditions for algebra equalities.

Finite-dimensionality characterized for Nichols algebras of diagonal type.

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.