Coboundary Lie bialgebras and commutative subalgebras of universal enveloping algebras

Benjamin Enriquez and Gilles Halbout

Vol. 229 (2007), No. 1, 161–184

DOI: 10.2140/pjm.2007.229.161

Abstract

We solve a functional version of the problem of twist quantization of a coboundary Lie bialgebra (g,r,Z). We derive from this that the formal Poisson manifolds g^∗ and G^∗ are isomorphic, and we construct an injective algebra morphism S(g^∗)^g↪U(g^∗). When (g,r,Z) can be quantized, we construct a deformation of this morphism. In the particular case when g is quasitriangular and nondegenerate, we compare our construction with Semenov-Tian-Shansky’s construction of a commutative subalgebra of U(g^∗). We also show that the canonical derivation of the function ring of G^∗ is Hamiltonian.

Keywords

coboundary Lie bialgebras, commuting families, linearization, Poisson algebras

Mathematical Subject Classification 2000

Primary: 17B63

Secondary: 17B80, 16W30

Milestones

Received: 4 April 2005

Revised: 23 November 2005

Accepted: 26 January 2006

Published: 1 January 2007