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
${\mathfrak b}^*$, the dual of ${\mathfrak b}$, is equipped with the coadjoint action of ${\mathfrak b}$ and is considered as an abelian ideal of $I{\mathfrak b}$.
Given an automorphism $\gamma$ the extended Dynkin diagram of ${\mathfrak g}$,
we construct an automorphism of $I{\mathfrak b}$ of the same order.
In type A, these automorphisms were recently constructed by Dror
Bar-Natan and Roland Van Der Veen in \cite{BNVDV} (where $I{\mathfrak b}$ is
denoted by $I{\mathfrak u}_n$).
Their construction is by hand and they ask for an explanation: this
note fully answers the question.
Origine | Fichiers produits par l'(les) auteur(s) |
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