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

Abstract

Let g be a complex simple Lie algebra with a Borel subalgebra b. Consider the semidirect product Ib=b⋉b⁎, where the dual b⁎ of b is equipped with the coadjoint action of b and is considered as an abelian ideal of Ib. We describe the automorphism group Aut(Ib) of the Lie algebra Ib. In particular we prove that it contains the automorphism group of the extended Dynkin diagram of g. In type An, the dihedral subgroup was recently proved to be contained in Aut(Ib) by Dror Bar-Natan and Roland van der Veen in [1] (where Ib is denoted by Iun). Their construction is ad hoc and they asked for an explanation which is provided by this note. Let n denote the nilpotent radical of b. We obtain similar results for Ib‾=b⋉n⁎ that is both an Inönü-Wigner contraction of g and the quotient of Ib by its center.