On the automorphisms of the Drinfel'd double of a Borel Lie subalgebra

Abstract

Let $\mathfrak{g}$ be a complex simple Lie algebra with Borel subalgebra $\mathfrak{b}$. Consider the semidirect product $I\mathfrak{b}=\mathfrak{b}⋉{\mathfrak{b}}^{\ast }$, where the dual ${\mathfrak{b}}^{\ast }$ 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 $\mathrm{Aut}(I\mathfrak{b})$ 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 $\mathrm{Aut}(I\mathfrak{b})$ 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.

arXiv comment17 pages, 1 table. It was asserted in second version that an automorphism of a Kac-Moody algebra associated to an automorphism of the extended Dynkin diagram is C[t]-linear. In fact, we found no proof for this, but we provide some evidence in this direction