On the automorphisms of the Drinfel'd double of a Borel Lie subalgebra
Résumé
Let ${\mathfrak g}$ be a complex simple Lie algebra with Borel subalgebra ${\mathfrak b}$.
Consider the semidirect product $I{\mathfrak b}={\mathfrak b}\ltimes{\mathfrak b}^*$, where
the dual ${\mathfrak b}^*$ of ${\mathfrak b}$, is equipped with the coadjoint action of
${\mathfrak b}$ and is considered as an abelian ideal of $I{\mathfrak b}$.
We describe the automorphism group ${\operatorname{Aut}}(I{\mathfrak b})$ of the Lie algebra $I{\mathfrak b}$.
In particular we prove that it contains the automorphism group of the extended Dynkin diagram of ${\mathfrak b}$.
In type $A_n$, the dihedral subgroup was recently proved to be contained
in ${\operatorname{Aut}}(I{\mathfrak b})$ by Dror
Bar-Natan and Roland Van Der Veen in arXiv:2002.00697 (where $I{\mathfrak b}$ is
denoted by $I{\mathfrak u}_n$).
Their construction is handmade and they ask for an explanation: this
note fully answers the question.
Origine : Fichiers produits par l'(les) auteur(s)
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