Olav Gravir Imenes, HiOA: Illuminating non-commutative algebraic geometry by using input from electroweak theory.
Abstract: Let A be an associative k-algebra. A point in the non-commutative affine sheaf Sch(A), is a simple module ρ0 : A → Endk(V). Understanding the differentiable structure, or the dynamics, of Sch(A), requires the introduction of a non-commutative phase space functor, an injecitve homomorphism i : A → Ph(A), with a universal derivation d : A → Ph(A) in the category of A-algebras, together with the choice of a dynamical structure, A(σ), a quotient of Ph(A) containing A and with a derivation δ, extending d.
A representation ρ0 : A(σ) → Endk(V) corresponds to a measurement of the parameters of A together with a tangent of A(σ) at the point ρ0. The moduli space of isomorphism classes, Simp(A(σ)), of finite dimensional (simple modules), ρ : A(σ) → Endk(V ), is therefore a kind of phase space of Sch(A). The choice of σ leads to a specific derivation, δ of A(σ), acting as a vector field [δ] on the moduli space Simp(A(σ)), a kind of a time-evolution.
In this talk we will use a method of modeling physics, due to Laudal (Geometry of Time-Spaces, World Scientific, 2011), where the generators of the algebra A(σ) represent the parameters of the physical phenomena we are interested in. We will be inspired by electroweak theory and shall therefore consider algebras generated by position, momentum, electric charge and weak charge operators. We will have to include infinite dimensional representations, and specialise to stable points. Borrowing notions and examples from physics give us a better understanding of parts of non-commutative algebraic geometry, and in some cases a better understanding of physics. But borrowing notions from physics, and comparing the resulting model to physics, is obviously not the same as doing physics, even though, in some cases, it looks remarkably similar.