# Algebraic structures connecting Lie theory and many body Physics

Lead Research Organisation:
University of Leeds

Department Name: Pure Mathematics

### Abstract

Physical Science can be thought of an attempt to understand, and make use of, the physical world. Part of the process of understanding any system is to make models of it. Such models can range from scaled-down versions, to mathematical simulations. This research proposal can be considered, from one motivating perspective, to be concerned with the mathematics needed in making useful mathematical models. (It is concerned, in particular, with the Algebra needed.)

Modelling as an approach to understanding is not limited to physics. One can attempt to investigate abstract structures (such as groups of symmetries) by modeling. In this setting the process is called Representation Theory.

We often find ourselves in the situation that a mathematical model of a physical system is still rather complicated, and itself benefits from modelling. Thus one aspect of this research proposal is concerned with the representation theory of structures which in turn model phenomena in physics (typically in Statistical Mechanics).

A wonderful feature of this modelling chain is that the non-abstract end system is amenable to other approaches, such as experiment and physical intuition. Thus not only does the representation theory inform the physics, but also the physics informs the representation theory. By pursuing this line, many exciting new pieces of mathematics have been discovered. And many more remain to be discovered.

Another great feature of this area of research is that mathematical models can be pushed into new generalisations and variations not obviously or intuitively indicated by (or easily available in) the physical context. This nicely underwrites the risks of research in an area where not every line of investigation is obvious or intuitive, even among those that do turn out to be physically important. In short, the methodology gives a way of sourcing radical new ideas in physical science.(In this specific proposal, however, the emphasis is primarily on the intrinsic interest of the mathematics, and on the contribution of physics `feeding back' into the study of challenging open problems in representation theory itself; in tandem with several proven technical devices from abstract algebra itself.)

In summary then, this research proposal is concerned with the representation theory of structures (such as groups and algebras) used, or potentially used, in physical modelling. The strategy is to use a tight interplay between the (very distinctive) mathematical and physical contexts.

Modelling as an approach to understanding is not limited to physics. One can attempt to investigate abstract structures (such as groups of symmetries) by modeling. In this setting the process is called Representation Theory.

We often find ourselves in the situation that a mathematical model of a physical system is still rather complicated, and itself benefits from modelling. Thus one aspect of this research proposal is concerned with the representation theory of structures which in turn model phenomena in physics (typically in Statistical Mechanics).

A wonderful feature of this modelling chain is that the non-abstract end system is amenable to other approaches, such as experiment and physical intuition. Thus not only does the representation theory inform the physics, but also the physics informs the representation theory. By pursuing this line, many exciting new pieces of mathematics have been discovered. And many more remain to be discovered.

Another great feature of this area of research is that mathematical models can be pushed into new generalisations and variations not obviously or intuitively indicated by (or easily available in) the physical context. This nicely underwrites the risks of research in an area where not every line of investigation is obvious or intuitive, even among those that do turn out to be physically important. In short, the methodology gives a way of sourcing radical new ideas in physical science.(In this specific proposal, however, the emphasis is primarily on the intrinsic interest of the mathematics, and on the contribution of physics `feeding back' into the study of challenging open problems in representation theory itself; in tandem with several proven technical devices from abstract algebra itself.)

In summary then, this research proposal is concerned with the representation theory of structures (such as groups and algebras) used, or potentially used, in physical modelling. The strategy is to use a tight interplay between the (very distinctive) mathematical and physical contexts.

### Planned Impact

The project, ``Algebraic structures connecting Lie theory and many body Physics'' has three main pathways for impact.

Firstly, this project has intrinsic academic impact due to its interdisciplinary nature in combining representation theory and statistical mechanics, and should provoke more collaboration and cross fertilisation between the two fields. For example, our research will lead to new models in physics such as allowing for a broader range of physical boundary conditions, leading to new applications in this area involving XXZ spin-chains and other such models.

Secondly, there will be impact on the landscape of mathematics in the UK and around the world. There are many academic stakeholders interested in the results that this project will generate. Including algebraists, geometers and mathematical physicists.

Thirdly, this project has potential educational impact through the likelihood that it will generate a new graduate course on this topic, of benefit to any institution that is part of the MAGIC network. It will also make the PDRA a valuable member of the UK scientific community, by training them in several related areas on modern research and in some rather unique and valuable methods of passing information between these areas.

Intrinsic to achieving the above impacts is our dissemination strategy. Apart from the usual dissemination via publication in internationally renowned journals and a commitment to open access via making preprints available on the arXiv, we have 4 other components to dissemination.

We will run an international workshop on this topic and related topics, the third in a series of very successful workshops partly funded by EPSRC. We will also attend conferences ourselves and well as give seminars around the UK and abroad. We have several ongoing research collaborations, these together with research visits and new collaborations will help dissemination. An enhanced website at the University of Leeds will be dedicated to the dissemination of the research outcomes of this project which will also facilitate more general outreach. Public outreach will also be achieved by PhD open days, organising of the Royal Institution Mathematics Masterclasses and talks in schools. We also currently supervise several PhD students in this area and the Co-I has supervised EPSRC summer projects.

Firstly, this project has intrinsic academic impact due to its interdisciplinary nature in combining representation theory and statistical mechanics, and should provoke more collaboration and cross fertilisation between the two fields. For example, our research will lead to new models in physics such as allowing for a broader range of physical boundary conditions, leading to new applications in this area involving XXZ spin-chains and other such models.

Secondly, there will be impact on the landscape of mathematics in the UK and around the world. There are many academic stakeholders interested in the results that this project will generate. Including algebraists, geometers and mathematical physicists.

Thirdly, this project has potential educational impact through the likelihood that it will generate a new graduate course on this topic, of benefit to any institution that is part of the MAGIC network. It will also make the PDRA a valuable member of the UK scientific community, by training them in several related areas on modern research and in some rather unique and valuable methods of passing information between these areas.

Intrinsic to achieving the above impacts is our dissemination strategy. Apart from the usual dissemination via publication in internationally renowned journals and a commitment to open access via making preprints available on the arXiv, we have 4 other components to dissemination.

We will run an international workshop on this topic and related topics, the third in a series of very successful workshops partly funded by EPSRC. We will also attend conferences ourselves and well as give seminars around the UK and abroad. We have several ongoing research collaborations, these together with research visits and new collaborations will help dissemination. An enhanced website at the University of Leeds will be dedicated to the dissemination of the research outcomes of this project which will also facilitate more general outreach. Public outreach will also be achieved by PhD open days, organising of the Royal Institution Mathematics Masterclasses and talks in schools. We also currently supervise several PhD students in this area and the Co-I has supervised EPSRC summer projects.

### Organisations

## People |
## ORCID iD |

Alison Parker (Principal Investigator) | |

Paul Martin (Co-Investigator) |

### Publications

Green Richard
(2017)

*On quasi-heredity and cell module homomorphisms in the symplectic blob algebra*in arXiv e-prints
Hazi A
(2021)

*Indecomposable tilting modules for the blob algebra*in Journal of Algebra
Hazi Amit
(2018)

*Indecomposable tilting modules for the blob algebra*in arXiv e-prints
King O
(2018)

*On central idempotents in the Brauer algebra*in Journal of Algebra
King Oliver
(2016)

*On central idempotents in the Brauer algebra*in arXiv e-prints
King Oliver H.
(2016)

*Decomposition matrices and blocks for the symplectic blob algebra over the complex field*in arXiv e-printsDescription | We have investigated the symplectic blob algebra and found new results about its structure. In particular we have determined the block structure in many cases using homomorphisms constructed by us. We have investigated the Brauer algebra and found new central idempotents for it. A paper on this topic appeared in 2018. We also investigated the partition algebra and a generalisation called the d-tonal partition algebra. Work is ongoing in collaboration with Ahmed, Benkart, King, Martin and Parker. We are currently investigating tilting modules for the blob algebra via KLR algebras jointly with our new postdoc Hazi. We have a preprint on the arXiv that determines tilting modules for the blob algebra in zero characteristic. This paper has now appeared in the Journal of Algebra |

Exploitation Route | We expect that our current results will be used by ourselves and others to further investigate these objects. |

Sectors | Other |