Representation theory of modular Lie algebras and superalgebras

Representation theory of Lie groups and Lie algebras has been a topic at the heart of mathematics for over 100 years with wide-ranging applications in mathematics and physics. This subject has origins in the view of Felix Klein in the 19th century that geometry of spacetime should be governed by its group of symmetries and the subsequent pioneering work of Sophus Lie to develop a theory of symmetries for differential equations.

Lie groups can be viewed as continuous symmetries of geometric objects. For example, a circle has infinitely many symmetries, namely rotations and reflections, which we can vary in a continuous way. Taking a step back we are able to view a Lie group more abstractly, and then representation theory provides the language to understand the different ways that a Lie group can act as symmetries. The Lie algebra of a Lie group is a first order approximation of a Lie group, which is more accessible to study, but retains all the local structure of the group. The abundance of continuous symmetry in mathematics and physics explains the wide ranging applications of this theory.

In the 1950s the "analytic theory" of Lie groups and Lie algebras was extended so that it can approached more algebraically, and this spurned a large area of mathematics now known as algebraic Lie theory. This is one of the most active areas of mathematics research today, which finds diverse applications across the physical sciences. An important area of algebraic Lie theory is the representation theory of modular Lie algebras. These Lie algebras can be thought of as versions of real or complex Lie algebras where usual arithmetic using real or complex numbers is replaced by modular arithmetic as is used in coding theory and cryptography.

The aim of this project is to exploit exciting recent developments in algebraic Lie theory to give a new perspective of the representation theory of modular Lie algebras. In order to understand representations of Lie algebras, we want to associate numerical data, which governs the structure of the representations. The most important pieces of data are the dimension and characters, and the ambitious goal of this project is to develop a methods for determining formulae for these.

The fundamental problems being tackled in the proposed research are of major international interest. Consequently, in the short term this research will make a substantial contribution to the competitiveness of the UK in modern methods in representation theory and Lie theory. The 2010 International Review of Mathematics states that Lie-Theoretic Representation Theory is "a blue-riband area of mathematics and the UK has great strength here". Further, the importance of the UK maintaining its strength and leadership in representation theory and noncommutative algebra is recognised in the EPSRC strategic focus.

The representation theory of Lie algebras serves as a prototype for other Lie theoretic algebras, for example rational Cherednik algebras. Therefore, the research is very likely to influence researchers working in the representation theory of related algebras. In addition the proposed research will introduce new combinatorial and geometric techniques, which are very likely to impact in the other areas of mathematics.

There is potential for major impact within mathematics and physics due to the connections with a range of areas of fundamental importance. For example, to the theory of quantum groups and Yangians, which is a very active area of research with applications in statistical physics and many other areas of physics; further, via the connections to affine W-algebras, there is the possibility of significant impact in conformal field theory and specifically the AGT conjecture. Such impact is likely to materialise in the medium to long term.

Representation theory more broadly has found many striking applications. For example, representations of finite groups have applications in studying molecular vibrations in chemistry, whereas representations of Lie groups are of central importance in quantum mechanics and in particle physics, for instance in the classification of elementary particles. There have also been many recent developments in the theory of categorification with applications to invariants of knots and links, and consequently to biology, chemistry and physics, for example to modelling DNA. Further applications, include the representation theory of symmetric groups in probability theory, which in turn had impact in the design of shuffling machines used in casinos. Further practical applications of representation theory will emerge in the future and the results of the proposed research have the potential to be a significant part of the input into these applications.

To summarize, this project forms part of a highly interconnected collection of research spanning modern mathematics, with applications across the physical sciences. Predicting which parts of this web of knowledge will have impact and quantifying this cannot be done with any certainty. As the proposed research is deeply connected to many important parts of this network of research, it will influence diverse areas in mathematics and the physical sciences. In the long term this may translate to significant scientific breakthroughs, and to impact on the economy and society more broadly.