On the geometry of Kähler-Poisson structures

Abstract

We prove that the Riemannian geometry of almost Kähler manifolds can be expressed in terms of the Poisson algebra of smooth functions on the manifold. Subsequently, Kähler-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.

References in corpus (5)

• Curvature and Gravity Actions for Matrix Models II: the case of general Poisson structure
• Discrete curvature and the Gauss-Bonnet theorem
• Representation theory of C-algebras for a higher order class of spheres and tori
• On the classical geometry of embedded surfaces in terms of Poisson brackets
• On the classical geometry of embedded manifolds in terms of Nambu brackets

Cited by in corpus (3)

• On Poisson geometries related to noncommutative emergent gravity
• Curvature and geometric modules of noncommutative spheres and tori
• On the geometry of compatible Poisson and Riemannian structures