Loops in the group of Hamiltonian symplectomorphisms
by
Andrés Viña
Departamento de Física. Universidad de Oviedo

Let (M, \omega) be a symplectic manifold; the group of Hamiltonian symplectomorphisms of M is denoted by Ham(M). If \psi = {\psi_t }_{t in [0, 1]} is a loop in Ham(M) at id, we define

\kappa(\psi)=exp æ è 2\pii ó õ S  \omega-2\pii ó õ 1 0  ft(\psit(q))dt ö ø ,

where f_t is a ``time" dependent Hamiltonian which defines \psi, and S is any surface bounded by the closed curve {\psi_t(q) }. We prove that \kappa(\psi) is independent of the point q in M. The variation of \kappa with \psi defines a closed Z-valued 1-form \Omega on the space of loops in Ham(M). It turns out that the values of \Omega are the degrees of maps from S¹ into S¹; moreover the form \Omega allows us to define a grading on \pi₂(Ham(M)).

When M is a coadjoint orbit of a semisimple Lie group G, the invariant \kappa(\psi) can be expressed in terms of the character of an irreducible representation of G.

Date received: January 29, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafw-09.