The term deformation quantization was introduced by Moshe Flato, who suggested that “any nontrivial associative deformation of an algebra of functions should be interpreted as a kind of `quantization' ”.
Deformation quantization is formally defined as the study of associative
-products of the form
, where
The concept is intensely studied by physical mathematicians, and has been intensively developed in recent years in the context of smooth Poisson manifolds, perhaps because of its potential applications in theoretical physics. Thus, it would be natural to consider such deformations `in the direction of Poisson brackets' by choosing
(cf. Drinfel'd), which is naturally antisymmetric. However, independently of the symplectic structure, one can consider more general deformations than that defined above, because the antisymmetry of
An especially interesting study is concerned with the deformation quantization on varieties with singularities. The cohomological implications of such singularities should lead to some very interesting mathematical properties and also to novel mathematical results.
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