  These notes present an introduction to an analytic version of deformation
quantization. The central point is to study algebras of physical observables
and their irreducible representations. In classical mechanics one deals with
real Poisson algebras, whereas in quantum mechanics the observables have the
structure of a real non-associative Jordan-Lie algebra. The non-associativity
is proportional to $\pl^2$, hence for $\pl\raw 0$ one recovers a real Poisson
algebra. This observation lies at the basis of `strict' deformation
quantization, where one deforms a given Poisson algebra into a $C^*$-algebra,
in such a way that the basic algebraic structures are preserved. Our main
interest lies in degenerate Poisson algebras and their quantization by
non-simple Jordan-Lie algebras. The traditional symplectic manifolds of
classical mechanics, and their quantum counterparts (Hilbert spaces and
operator algebras which act irreducibly) emerge from a generalized
representation theory. This two-step procedure sheds considerable light on the
subject. We discuss a large class of examples, in which the Poisson algebra
canonically associated to an (integrable) Lie algebroid is deformed into the
Jordan-Lie algebra of the corresponding Lie groupoid. A special case of this
construction, which involves the gauge groupoid of a principal fibre bundle,
describes the classical and quantum mechanics of a particle moving in an
external gravitational and Yang-Mills field.
