Universal deformation formulae, symplectic Lie groups and symmetric spaces

Pierre Bieliavsky, Philippe Bonneau and Yoshiaki Maeda

Vol. 230 (2007), No. 1, 41–57

DOI: 10.2140/pjm.2007.230.41

Abstract

We define a class of symplectic Lie groups associated with solvable symmetric spaces. We give a universal strict deformation formula for every proper action of such a group on a smooth manifold. We define a functional space where performing an asymptotic expansion of the nonformal deformed product in powers of the deformation parameter yields an associative formal star product on the symplectic Lie group at hand. The cochains of the star product are explicitly given (without recursion) in the two-dimensional case of the affine group ax + b. The latter differs from the Giaquinto–Zhang construction, as shown by analyzing the invariance groups. In a Hopf algebra context, the above formal star product is shown to be a smash product and a compatible coproduct is constructed.

Keywords

universal deformation formula, symplectic Lie group, symmetric space, Lie group actions

Mathematical Subject Classification 2000

Primary: 54C40, 14E20

Secondary: 46E25, 20C20

Milestones

Received: 25 March 2005

Accepted: 3 July 2006

Published: 1 March 2007