Noncommutative tangent bundle: The Phase Space

Abstract

In this thesis we study thwe phase space Ph(A) for an associative algebra A, with k algebraically closed (when needed we will assume k=C). The phase space has a universal property analogous to that of the module of Kähler differentials in classical algebraic geometry, and for this and other reasons it can be regarded as a kind of non-commutative (co)tangent bundle. In particular, we include a result showing that with A commutative and smooth, the commutativized version Ph(A)_com of Ph(A) will be 'locally trivial'. We also define a cohomology theory for Ph(A) and use it to prove an algebraic variant of an 'inverse function theorem'. Finally we look at representations of the phase space, and how they can be interpreted geometrically.