Friday, November 16, 2012

1103.5862 (Joakim Arnlind et al.)

On the geometry of Kähler-Poisson structures    [PDF]

Joakim Arnlind, Gerhard Huisken
We prove that the Riemannian geometry of almost K\"ahler manifolds can be expressed in terms of the Poisson algebra of smooth functions on the manifold. Subsequently, K\"ahler-Poisson algebras are introduced, and it is shown that a corresponding purely algebraic theory of geometry and curvature can be developed. As an illustration of the new concepts we give an algebraic proof of the statement that a bound on the (algebraic) Ricci curvature induces a bound on the eigenvalues of the (algebraic) Laplace operator, in analogy with the well-known theorem in Riemannian geometry. As the correspondence between Poisson brackets of smooth functions and commutators of operators lies at the heart of quantization, a purely Poisson algebraic proof of, for instance, such a "Gap Theorem", might lead to an understanding of spectral properties in a corresponding quantum mechanical system.
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