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 .

Key Words:

• Lie bialgebra
• Parabolic subalgebra
• Drinfeld–Jimbo r-matrix
• Classical double

Mathematics Subject Classification:

• 17B62
• 17B20

Acknowledgment

The author is thankful to Professor A. Stolin who suggested the problem and gave valuable ideas.

Notes

^#Communicated by Zelmanov.