Representation Theory

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A group is a fundamental mathematical notion that developed over time from fairly concrete settings into an object that is now abstractly defined purely in terms of axioms. In the realm of all groups, there are large reservoirs of concrete and natural examples that can be studied and understood far more easily than general abstract groups. Linear groups of transformations are of this kind. Group representation theory aims at understanding the various ways in which groups can be realized as linear groups. An analogous situation occurs for algebras, which are basic mathematical objects as well. There are various types of them, and I am interested in algebras of Lie type, which are among the most widely studied because of their applications, relevance and inner beauty. Abstract Lie algebras can be made concrete by means representations. These produce mirror images (not all of them entirely faithful) of the former in the shape of Lie algebras of linear transformations, which are among the most concrete types of Lie algebras. Representations are usually made up by combining unbreakable or indecomposable components. One of the goals of the representation theory of Lie algebras (resp. groups) is to construct, understand and classify all indecomposable representations for each and every one the various types of Lie algebras (resp. groups). The most well understood and better behaved types of Lie algebras are called semisimple. Their inner structure and indecomposable representations are well known. However, most Lie algebras other than semisimple have a wild representation theory and is exceedingly difficult, almost utopian, to classify all indecomposable representations for them, regardless of how elementary the Lie algebra itself might be. Nevertheless, I have identified a large but suitable class of non-semisimple Lie algebras for which the classification of certain types of indecomposable modules is feasible, which is what I plan to do. Significant progress in this direction will greatly augment our present knowledge of the representation theory of these wildly behaved Lie algebras, and build structure and foundation to an area where our body of knowledge is not yet well organized. In the case of groups, I am focused on a distinguished kind of representation, discovered by R. Steinberg, that plays a prominent role in the representation theory of an important class groups called finite groups of Lie type. Group representations come into two types, ordinary and modular, the latter obtained from the former by means of a reduction process. I am trying to locate and identify the indecomposable components of the modular Steinberg representation. Due to the importance of this representation in group theory, the solution to this problem is likely to attract interest and attention. An even greater impact of my research, outside of the mathematical community, is to the roughly 1000 students that I will teach over the next 5 years. Funding for my research will allow me to maintain a reasonable flow of advanced undergraduate students, graduate students and postdoctoral fellows, with whom to share and discuss my research projects. While I am teaching the hundreds of beginning undergraduate students, they will greatly benefit from having a professor that not only knows the subject material, but is actively engaged in research. The young minds that come to our classroom need to be nurtured, challenged and led to grow to their full potential, and it is clear to me that this can only be achieved with the help of those that are most active in their field of expertise. Canada will benefit by allocating resources so that its youth receive top quality education.