Operad filtrations and quantization

Abstract: The celebrated problem of deformation quantization discusses deformations of Poisson algebras into associative algebras, a question that is, in the end, motivated by quantum mechanics. I shall discuss this question and some of its generalisations from the purely algebraic point of view using the theory of operads. In particular, I shall show how to prove that there are, in a strict mathematical sense, only two meaningful deformation problems for Poisson algebras, namely deforming them in the class of all Poisson algebras or all associative algebras, and there is only one meaningful deformation problem for the so called almost Poisson algebras (also sometimes known as generic Poisson algebras), namely deforming them in the class of all almost Poisson algebras. For instance, this explains the existing body of work in the mathematical physics literature asserting that some classes of non-associative star products cannot be alternative, are always flexible etc.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar