Kähler-Poisson Algebras

Abstract

In this thesis, we introduce Kähler-Poisson algebras and study their basic properties.The motivation comes from differential geometry, where one can show that the Riemannian geometry of an almost Kähler manifold can be formulated in terms of the Poisson algebra of smooth functions on the manifold.It turns out that one can identify an algebraic condition in the Poisson algebra (together with a metric) implying that most geometric objects can be given a purely algebraic formulation.This leads to the definition of a Kähler-Poisson algebra, which consists of a Poisson algebra and a metric fulfilling an algebraic condition.We show that every Kähler-Poisson algebra admits a unique Levi-Civita connection on its module of inner derivations and, furthermore, that the corresponding curvature operator has all the classical symmetries.Moreover, we present a construction procedure which allows one to associate a Kähler-Poisson algebra to a large class of Poisson algebras.From a more algebraic perspective, we introduce basic notions, such as morphisms and subalgebras, as well as direct sums and tensor products.Finally, we initiate a study of the moduli space of Kähler-Poisson algebras; i.e for a given Poisson algebra, one considers classes of metrics giving rise to non-isomorphic Kähler-Poisson algebras.As it turns out, even the simple case of a Poisson algebra generated by two variables gives rise to a nontrivial classification problem.

Locations

• Linköping studies in science and technology. Dissertations - View - PDF
• KTH Publication Database DiVA (KTH Royal Institute of Technology) - View - PDF