Two-forms and Noncommutative Hamiltonian dynamics

Abstract

In this paper we extend the standard differential geometric theory of Hamiltonian dynamics to noncommutative spaces, beginning with symplectic forms. Derivations on the algebra are used instead of vector fields, and interior products and Lie derivatives with respect to derivations are discussed. Then the Poisson bracket of certain algebra elements can be defined by a choice of closed 2-form. Examples are given using the noncommutative torus, the Cuntz algebra, the algebra of matrices, and the algebra of matrix valued functions on $\Bbb R^2$.