Hamilton Araujo (University of Haute Alsace in Mulhouse)
This talk consists of a discussion around the analytical localization of the algebras of functions in deformation quantization (a theory that uses algebraic deformation to quantify the algebras of commutative functions for describe some aspects of quantum mechanics in physics) i.e. functions that are only defined on a open set of the variety, compared with the open non-commutative localization in the algebra of the deformed functions. The most elementary example of localization is the passage of the ring of integers to rational numbers (fractions), where some numbers are made invertible. We first describe the algebraic framework of the algebraic localization: there is a general process that is not very explicit, and there is the construction of g: Ore which is much more concrete, but is more particular because the multiplicative set (the set of future
denominators) must respect some condition (Ore conditions). Then we look at two basic examples: functions that are defined on an open sets and germs of functions in a point.
In the first case we obtain the equivalence between the analytical and the algebraic approach. The main tool of the demonstration is based on the analytical work of Whitney, Malgrange and especially on the book of J.-C. Tougeron. For germs, we obtain also some equivalence, but the choice -the most natural one- fails, and we must slightly modify the multiplicative set. At the end we discuss a more general and algebraic framework that allows us to formulate a question "Are localization and deformation commutative?".
A non-Ore example is also given. The talk is based on a preprint with B.Hurle and M.Bordemann.
-Preprint: Noncommutative localization in smooth deformation quantization. ArXiv: arxiv.org/abs/2010.15701. 2020.
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