Edinburgh Hodge Institute

In the paper 'arXiv:1410.5207' (to appear in Journal of Noncommutative Geometry) Michel Van den Bergh and I generalize some classical birational transformations to the noncommutative setting. In particular we show that 3-dimensional quadratic Sklyanin algebras (noncommutative projective planes) and 3-dimensional cubic Sklyanin algebras (non-commutative quadrics) have the same function field. In the same vein we construct an analogue of the Cremona transform for noncommutative projective planes.

In this talk I will explain the approach and motivation for our paper. The motivation is based on the construction of the Cremona transform (in the commutative setting) using linear systems. I will also recapitulate some relevant facts about noncommutative surfaces and Artin-Schelter regular (Z-)algebras.

Quantum nonlocality is the strange phenomenon where two entangled particles can affect each other instantaneously, regardless of how far apart they are. I will explain how this phenomenon can be described geometrically as a nonnoetherian singularity in spacetime. I will then introduce the notion of a noncommutative resolution of such a singularity, and describe how this notion fits into the quantum world.

This is joint work with Michel Van den Bergh. We generalize standard results about non-commutative resolutions of quotient singularities for finite groups to arbitrary reductive groups. We show in particular that quotient singularities for reductive groups always have non-commutative resolutions in an appropriate sense.

We discuss a number of examples, both new and old, that can be treated using our methods; twisted non-commutative crepant resolutions exist in previously unknown cases for determinantal varieties of symmetric and skew-symme- tric matrices.

We describe Lie--Rinehart algebras in the tensor category of linear maps in the sense of Loday and Pirashvili and construct a functor from Lie-Rinehart algebras (in the category of linear maps) to Leibniz algebroids.

The construction of the universal R-matrix for quantum groups produces solutions of the Yang-Baxter equation on tensor products of representations of that quantum group. This gives an action of the braid group of type A, endowing the category of finite dimensional Uq(g)-representations with a structure of a braided tensor category.

I will explain how the theory of quantum symmetric pairs allows an analogous construction of a universal K-
matrix, which produces solutions of the reflection equation on tensor products of representations of that quantum group. This gives a representation of the braid group of type B, endowing the category of finite dimensional Uq(g)-representations with a structure of a braided tensor category with a cylinder twist.

In my talk I will construct a functor from some class of skew fields to some class of commutative fields with map \( Q( C[x, \partial_x]) \to Q (C[x, y]) \) as the main example. I will also provide some link between this construction and some known results and conjectures on skew fields.