Power sum elements in the $G_2$ skein algebra
We study the skein algebras of surfaces associated to the exceptional Lie group $G_2,$ using Kuperberg webs. We identify two 2-variable polynomials, $P_n(x,y)$ and $Q_n(x,y),$ and use threading operations along knots to construct a family of central elements in the $G_2$ skein algebra of a surface,...
Gespeichert in:
Hauptverfasser: | , , , , , |
---|---|
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
Online-Zugang: | Volltext bestellen |
Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
Zusammenfassung: | We study the skein algebras of surfaces associated to the exceptional Lie
group $G_2,$ using Kuperberg webs. We identify two 2-variable polynomials,
$P_n(x,y)$ and $Q_n(x,y),$ and use threading operations along knots to
construct a family of central elements in the $G_2$ skein algebra of a surface,
$\mathcal{S}_q^{G_2}(\Sigma),$ when the quantum parameter $q$ is a
$2n\text{-th}$ root of unity. We verify these elements are central using
elementary skein-theoretic arguments. We also obtain a result about the
uniqueness of the so-called transparent polynomials $P_n$ and $Q_n.$ Our
methods involve a detailed study of the skein modules of the annulus and the
twice-marked annulus. |
---|---|
DOI: | 10.48550/arxiv.2310.01773 |