Poisson Lie groups and pentagonal transformations

✍ Scribed by S. Zakrzewski

• Publisher: Springer
• Year: 1992
• Tongue: English
• Weight: 286 KB
• Volume: 24
• Category: Article
• ISSN: 0377-9017
• DOI: 10.1007/bf00429998

✦ Synopsis

Symplectic pentagonal transformations are intimately related to global versions of Poisson Lie groups (Manin groups, S*-groups, or symplectic pseudogroups). Symplectic pentagonal transformations of cotangent bundles, preserving the natural polarization, are shown to be in one to one correspondence with pentagonal transformations of the base manifold with a cocycle (if the base is connected and simply connected). By the results of Baaj and Skandalis, this allows to quantize (at the C*-algebra level!) those Poisson Lie groups, whose associated symplectic pentagonal transformation admits an invariant polarization. The (2n)2-parameter family of Poisson deformations of the (2n + 1)-dimensional Heisenberg group described by Szymczak and Zakrzewski is shown to fall into this case.