Indecomposable Lie algebra representations

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Lie algebras attract much attention due to their intrinsic beauty, importance, and deep connections to other areas of Mathematics and Mathematical Physics, such as differential geometry, particle physics, quantum groups, differential equations, simple groups of Lie type, etc. As is the case with most mathematical objects, Lie algebras are defined axiomatically. In addition to the initial objects that suggested the given axioms, many more suddenly spring to life, and one of the goals of the theory, albeit utopian, is to classify all possible Lie algebras by identifying as one all those that are exact mirror images of each other. Of great aid in this endeavor is Representation Theory, which studies all images (faithful or not) of a given Lie algebra; these consist of the various ways in which an abstract Lie algebra can act as a concrete Lie algebra of linear transformations of a vector space. There is a well behaved family of Lie algebras, called semisimple, whose (finite dimensional) representation theory is well understood in terms of atomic or irreducible representations, which can be combined to produce all others. However, the vast majority of Lie algebras are not semisimple and the atomic representations, now called indecomposable, are extremely difficult to understand. These are the ones I propose to investigate, building upon my prior work on the subject, while aiming deeper and further. Their understanding would constitute a significant contribution to our knowledge of the representation theory of Lie algebras in general, beyond the classical case of semisimple Lie algebras, and in area where not much is known at present. My research program will have a direct impact on the graduate and undergraduate university students I teach, including prospective teachers, as well as on the high school and elementary students with whom I interact through outreach activities. By being close to the frontier of knowledge, constantly pondering and solving research questions, and by working collaboratively with other active investigators, I am in a position to inspire and stimulate the future generation of critical thinkers and creators and help them reach their full potential. This can only be achieved by those instructors who are most active in their field of expertise. Canada will benefit by allocating resources so that its youth receive a top quality education.