Integrability of invariant metrics on the Virasoro group

Boris Kolev (1, 2) , Adrian Constantin (3) , Jonatan Lenells (3)

1 I2M - Institut de Mathématiques de Marseille
2 LATP - Laboratoire d'Analyse, Topologie, Probabilités
3 CMS - Center for Mathematical Sciences

Each right-invariant metric on the Virasoro group induces a Hamiltonian vector field on the dual of the Lie algebra of the Virasoro algebra equipped with the canonical Lie-Poisson structure. We show that the Hamiltonian vector fields X_k induced by the metrics given at the identity by the H^k Sobolev inner products are bi-Hamiltonian relative to a modified Lie-Poisson structure only for k=0 and k=1.

Mots clés: Bi-Hamiltonian structures; Virasoro group

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]

Boris Kolev, Adrian Constantin, Jonatan Lenells. Integrability of invariant metrics on the Virasoro group. Physics Letters A, 2006, 350 (1-2), pp.75-80. ⟨10.1016/j.physleta.2005.10.011⟩. ⟨hal-00095908⟩