Abelian deformations

Certain physical problems lead to a need for quantization in a context where a Poisson bracket does not provide the direction. Nambu mechanics in a three-dimensional "phase space" is one example of this. Another is the problem of quantization on coadjoint orbits, especially, on singular orbits. Abelian (*)-products are governed by Harrison cohomology, which is often, but mistakenly, said to be trivial. In fact, varieties with singularities, including simple examples of physical relevance, do have a nontrivial Harrison cohomology. Moreover, Harrison cohomology is not always decisive. A Minkowski space is a smooth manifold with vanishing Harrison cohomology; the coordinate algebra. admits, nevertheless, nontrivial abelian deformations.