On the Classical Double of Parabolic Subalgebras#

Abstract

Given a complex simple finite-dimensional Lie algebra g with fixed root system, there exists a so-called classical Drinfeld–Jimbo r-matrix, r. Consider any parabolic subalgebra P S ⊆ g defined by a subset S of the set of simple roots. We prove that the Lie bialgebra structure on g defined by r can be restricted to P S . Moreover, it turns out that the corresponding classical double D(P S ) is isomorphic to g ⊕ Red(P S ), where Red(P S ) denotes the reductive part of P S .