Grant: Discovery Grants Program - Individual

Since their emergence in the work of the 19th century Norwegian mathematician Sophus Lie on continuous transformation groups, Lie algebras and Lie groups have become fundamental objects of research in mathematics. A wide range of problems in mathematics and physics can be formulated as questions about certain linear realizations of Lie algebras and Lie groups, also known as their representations. Numerous examples of these representations occur in a broad spectrum of domains of mathematical research beyond algebra, such as algebraic geometry, number theory, and mathematical physics, to name a few. The advent of supersymmetry in particle physics led to the introduction of a class of non-associative algebras, called Lie superalgebras, that can be described concretely as graded analogues of Lie algebras. The structure and representations of Lie superalgebras yield a theory that is just as rich and fascinating as the one for Lie algebras. However, Lie superalgebras also exhibit new phenomena that are not present in the Lie algebra setting. Furthermore, Lie superalgebras and more generally the machinery of superized algebraic structures can be utilized to unite seemingly different mathematical facts into one general assertion that reproduces the original ones by means of even and odd specializations. A key ingredient in the study of representations of Lie (super)algebras is understanding various algebras of invariants, and in particular algebras of invariant differential operators. The prototypical example of an algebra of differential operators is the Weyl algebra, or its odd analogue known as the Clifford algebra. The aforementioned algebraic structures have quantum deformations as well, leading to an important class of Hopf algebras called quantized enveloping (super)algebras, and their representations. In the quantized context, these representations constitute a category that has fascinating links with topology. In my research over the next five years, I will investigate questions about representations of Lie superalgebras and quantized enveloping (super)algebras with profound connections with geometry and combinatorics. My first goal is to obtain a deep understanding of the invariant theory in the superized and quantized contexts using tools from supergeometry, category theory, and Hopf algebras. My second goal is to generalize the powerful tools of classical representation theory and use them to unravel new connections between the field of Lie theory and the combinatorics of symmetric functions. This proposal offers a plethora of exciting and important research directions with significant impact in the fields of representation theory, combinatorics, and mathematical physics. The breadth and originality of the proposed research will generate excellent opportunities for the training of Highly Qualified Personnel at undergraduate, graduate, and postdoctoral levels, at the forefront of rapidly growing areas of mathematics.