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On Embedding Singular Poisson Spaces

  • Author(s): McMillan, Aaron Fraenkel
  • Advisor(s): Weinstein, Alan D
  • et al.

This dissertation investigates the problem of locally embedding singular Poisson spaces. Specifically, it seeks to understand when a singular symplectic quotient V/G of a symplectic vector space V by the linear symplectic action of a group G is realizable as a Poisson subspace of some Poisson manifold (R^n,{ , }).

The local embedding problem is recast in the language of schemes and reinterpreted as a problem of extending the Poisson bracket to infinitesimal neighborhoods of an embedded singular space. Such extensions of a Poisson bracket near a singular point p of V/G are then related to the cohomology and representation theory of the cotangent Lie algebra at p.

Using this framework, it is shown that the real 4-dimensional quotient V/Z_n (n odd, Z_n the cyclic group of order n) is not realizable as a Poisson subspace of any (R^(2n+6),{ , }), even though the underlying variety algebraically embeds into R^(2n+6). The proof of this nonembedding result hinges on a refinement of the Levi decomposition for Poisson manifolds to partially linearize any extension with respect to the Levi decomposition of the cotangent Lie algebra of V/G at the origin. Moreover, in the case n=3, this nonembedding result is complemented by a concrete realization of V/Z_3 as a Poisson subspace of R^78.

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