Algebraic deformation and tangent categories

Marcello Lanfranchi – 27 March 2024

In the algebraic approach to Quantum Mechanics, the observables of a system, i.e. the physical quantities one can measure of a system, are mathematically represented by a *-algebra, which is an associative Banach algebra with an involution map. Employing the celebrated theorem of Gelfand and Najmark one can show that commutative *-algebras formalize classical observables, while non-commutative ones represent the observables of a quantum system. In the algebraic approach, usually, one starts with a well-known classical theory and builds a non-commutative version of the corresponding commutative *-algebra. This technique of building noncommutative algebras from commutative ones is called deformation quantization and it is based on the principles of deformation theory. Deformation theory aims to construct algebraic objects of some type, e.g. associative algebras, starting from other ones, in a controlled way, so that one can restore the original objects by "killing" some extra terms. This construction, in my opinion, can be interpreted geometrically. In this talk, I would like to discuss some ideas and intuitions I developed to employ tangent category theory to interpret geometrically the deformation of an algebraic object. In particular, I will show that the category of operad is itself a tangent category, whose vector fields are related to infinitesimal deformations of the corresponding algebras.