Lie algebras and superalgebras: representations and structure theory

点击翻译按钮获取中文摘要

The idea of symmetry has played a central role in geometry, architecture, and art from ancient times. In the 1800's the notion of a group arose as the abstract algebraic structure encompassing all symmetries of a system. Towards the end of the 1800's, the Norwegian mathematician Sophus Lie disguised a class of groups which carries an additional structure reflecting the continuous structure of the system under consideration. These groups, called Lie groups, have become one of the most powerful tools in studying various geometric, analytic, and physical structures. Lie groups are highly non-linear but locally their structure is the same at every point. Thus, in many respects, studying a Lie group can be reduced to the simpler problem of studying a linear approximation, called a Lie algebra. The notion of a Lie superalgebra is a further generalization that traces its roots to the notion of supersymmetry in physics. Working with Z_2-graded objects, the mathematical terminology for “super-objects” in physics, is also the natural way to treat simultaneously commuting and anti-commuting qualities in mathematics. Thus, superalgebras provide the natural setting for problems arising in different field of mathematics, e.g. homological algebra. Lie algebras and Lie superalgebras - their structure and representation theory - have been studied extensively since the 1950's. Various classes of Lie (super)algebras and various classes of representations of Lie (super)algebras remain at the centre of research in Lie theory.The long-term goal of my research is two-fold: to study representations of infinite dimensional Lie algebras and to study the structure of Lie superalgebras and to obtain new results that unify and explain known phenomena for Lie algebras and Lie superalgebras. The current proposal is centred around four particular directions of research: 1. Weight representations of affine and finitary Lie algebras. 2. Left-symmetric superalgebras (LSSA). 3. Lagrangian subalgebras of simple Lie superalgebras. 4. Generalizations of root systems. I expect the outcomes of this research to have considerable impact in the areas of representation theory and combinatorics, as well as applications to physics. The proposal contains a number of projects suitable for training HQP at all levels. I intend to supervise and support three Ph.D. students, three M.Sc. students, five undergraduate students, and to co-supervise two postdoctoral fellows over the course of the grant.