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Pré-Publication, Document De Travail Année : 2020

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.
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Dates et versions

hal-02876579 , version 1 (21-06-2020)
hal-02876579 , version 2 (22-06-2020)

Identifiants

Citer

Michael Bulois, Nicolas Ressayre. On the automorphisms of the Drinfel'd double of a Borel Lie subalgebra. 2020. ⟨hal-02876579v1⟩
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