Reconstructing broken symmetry

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Consider the two-dimensional plane. What are its symmetries? We might think of rotations, reflections and translations, together with combinations of these. But might there be others? In fact, there are and we can be sure we have found them all by turning the question into a problem in algebra rather than geometry. By doing so, we find that every symmetry of the plane is a combinations of linear transformations (which include all of those mentioned above) and a further family of generalised (non-linear) shears.The translation into a problem in algebra is achieved by considering the so-called coordinate algebra of the geometric space. A symmetry of the space then precisely corresponds to an automorphism of the coordinate algebra. An automorphism of an algebra is a map from the algebra to itself that preserves the additive and multiplicative structure of the algebra and that has an inverse - just as a symmetry is a map from the space to itself which preserves geometric structure (e.g. angles and lengths) and is "undo-able". The set of automorphisms of an algebra forms a group under composition, so our original question is reformulated as one of describing the automorphism group of the coordinate algebra of our space.This is a classical problem - and is very hard in general. The example of the plane is rather misleading, as for three dimensions and above, the automorphism group has been proved to contain "wild" elements; that is, automorphisms that cannot be described in elementary terms as above.So this is not the problem we propose to address. Rather, our interests lie in the world of noncommutative algebraic geometry. (The coordinate algebras referred to above are in particular commutative algebras.) Here there is a well-known but not well understood phenomenon of symmetry breaking. Noncommutative or "quantum" spaces are usually more rigid than their classical commutative counterparts, in the sense that they have fewer symmetries. More precisely, noncommutative coordinate algebras typically have smaller automorphism groups.This leads naturally to the following question: where has the symmetry gone? The aim of this project is to provide an answer, showing that the "hidden" symmetries are recoverable as isomorphisms between different quantizations of the space. In technical language, we have an automorphism groupoid ("a group with many objects") that reduces to the original automorphism group in the classical limit. Constructing this groupoid, even for small examples, requires techniques from the spectrum of pure mathematics, includng noncommutative algebra, algebraic geometry and cohomology theory among others.Our goal is to fully understand this groupoid for certain quantizations. Specifically, we shall consider the plane, higher-dimensional affine spaces and some other carefully chosen examples. In doing so we shall develop general theory that can be applied to many further spaces and their quantizations.