Representation theory of modular Lie algebras and superalgebras

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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.