Optimal reduction of the cotangent bundle of a 2-step nilpotent Lie group

VERÓNICA S. DIAZ; MARÍA LAURA BARBERIS

Bariloche

Encuentro; Segundo Encuentro Iberoamericano sobre Geometría, Mecánica y Control, en honor de Hernán Cendra; 2011

Instituto Balseiro

Let G be a Lie group with Lie algebra g and T*G its cotangent bundle with symplectic structure w_{Sigma}, where w_{Sigma} is the sum of the canonical symplectic form and an invariant magnetic term, which depends on a Lie algebra two-cocycle Sigma:g x g -->R. It was shown in [OR] that the optimal reduced spaces of (T*G, w_{Sigma}) are symplectomorphic to the orbits of certain affine action on g*. We study the case when G is a 2-step nilpotent Lie group. In particular, starting from the Heisenberg group, we consider different examples of optimal reduced spaces, varying the two-cocycle Sigma. We obtain an explicit expression of the symplectic structure in these cases, and study conditions for having a Lie group structure on the reduced spaces. [OR] J.-P. ORTEGA, T.S. RATIU, The reduced spaces of a symplectic Lie group action, Annals of Global Analysis and Geometry 30(4), 335–38(2006).