# Reconstructing broken symmetry

Lead Research Organisation:
Lancaster University

Department Name: Mathematics and Statistics

### Abstract

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.

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.

### Planned Impact

Impact will be largely academic. The proposal is primarily concerned with the advancement of mathematical knowledge and the development of researchers in mathematics, specifically the Principal Investigator and the requested Research Associate.

It also aims to enhance what is already an area of strength for the UK, namely noncommutative algebra, by the investigation of a novel idea which will spark further activity and development. The core of the proposal is the development of a new way of looking at an old problem; the techniques we anticipate using will draw from across pure mathematics, with exchange of knowledge between practitioners from different areas being a vital component.

Direct economic and social impact from this work is only plausible after a significant period of time. We view our proposal as placed at the beginning of a relay: our work provides knowledge and tools for other mathematicians and mathematical physicists, which they in turn may exploit, leading to understanding that theoretical physicists working on unified quantum gravity theories might use, for example. Society has seen very significant and unexpected economic and societal impacts arising from work in fundamental physics on a number of occasions; in turn these were often ultimately dependent on underpinning mathematical work.

As part of the proposal we have indicated our keenness to communicate directly with the public. Rather than hide the mathematical underpinnings away, we wish to show how advanced mathematics is not beyond the public's appreciation and how, investigated for its own sake and inspired by its internal beauty, pure mathematics can ultimately lead to ideas and technology that change their lives.

It also aims to enhance what is already an area of strength for the UK, namely noncommutative algebra, by the investigation of a novel idea which will spark further activity and development. The core of the proposal is the development of a new way of looking at an old problem; the techniques we anticipate using will draw from across pure mathematics, with exchange of knowledge between practitioners from different areas being a vital component.

Direct economic and social impact from this work is only plausible after a significant period of time. We view our proposal as placed at the beginning of a relay: our work provides knowledge and tools for other mathematicians and mathematical physicists, which they in turn may exploit, leading to understanding that theoretical physicists working on unified quantum gravity theories might use, for example. Society has seen very significant and unexpected economic and societal impacts arising from work in fundamental physics on a number of occasions; in turn these were often ultimately dependent on underpinning mathematical work.

As part of the proposal we have indicated our keenness to communicate directly with the public. Rather than hide the mathematical underpinnings away, we wish to show how advanced mathematics is not beyond the public's appreciation and how, investigated for its own sake and inspired by its internal beauty, pure mathematics can ultimately lead to ideas and technology that change their lives.

## People |
## ORCID iD |

Jan Edward Grabowski (Principal Investigator) |

Description | We have succeeded in the main goal of the project, namely to find a formal mathematical framework to ask and answer questions on the symmetries of noncommutative spaces. Through engaging with the existing literature, we have identified the correct language to phrase our questions (an area of work known as noncommutative projective geometry) and we have defined an object to encode all of the symmetries. It is a groupoid - this is an object with many "points", each representing a different noncommutative space, and points are related precisely if the spaces they represent are related by a symmetry (technically, an isomorphism). The work on the project has made it clear that the right question to ask is, in this language, "how many points are connected to a chosen one?". Using some techniques from (usual, commutative) geometry, it is possible to make some estimates for some noncommutative spaces of special interest. |

Exploitation Route | We intend to continue working on this project as a collaboration between the PI and RA. |

Sectors | Other |