Research

Overview

Representation theory is the bridge between abstract algebra problems and linear algebra problems via the method of identifying abstract elements as linear functions in a compatible way. In effect, representation theory allows one to use extensive techniques and theorems of linear algebra to better understand algebraic objects; conversely, one can utilize the inherent structure of an algebraic object to obtain desirable linear functions via its representations.

The theory of representations is often applied to the domain of quantum algebra: the area of mathematics that in some way relates to the fact that noncommutativity is found in quantum mechanics. An important instance of noncommutative objects are known as quantum groups, whose representation theories are known to have particular importance in quantum physics. One such family of quantum groups are called Yangians, which come equipped with the foundational fact that their representations produce rational solutions to the quantum Yang-Baxter equation. The structure and representation theory of Yangians has become a study in and of itself, with such research expanding to super Yangians: a formulation of Yangians in terms of the mathematics of supersymmetry. However, the theory of super Yangians is comparatively less developed to its non-super counterpart, which is the cause of my PhD research.

Selected Presentations

  • Super Yangians: Where We Are Today

    • May 2022 | PIMS-USask Geometry, Algebra and Physics Seminar | University of Saskatchewan
    • Abstract: Given any finite-dimensional simple Lie algebra $\mathfrak{a}$ over $\mathbb{C}$, the Yangian $\mathbf{Y}(\mathfrak{a})$ is a certain unital associative $\mathbb{C}$-algebra. In particular, Yangians form a family of so-called quantum groups. The main property these algebras is the foundational fact that their representations produce rational solutions to the quantum Yang-Baxter equation. The structure and representation theory of Yangians has become a study in and of itself and has expanded to the study of super Yangians based on Lie superalgebras; however, the theory of super Yangians is comparatively less developed than its non-super counterpart. In this talk, we will survey what recent advancements have been made in the study of super Yangians and view what else remains to do.

  • A Friendly Introduction to Representation Theory

    • October 2021 | Mathematical and Statistical Sciences Graduate Colloquium | University of Alberta
    • Abstract: Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. When that abstract algebraic object is being represented on a finite-dimensional vector space, its elements are described by matrices and its algebraic operations are described by matrix multiplication and matrix addition. The power of such methodology allows one to reduce abstract algebra problems to linear algebra problems. In this talk, we will introduce the basic notions of representation theory with a focus on group representations.
    • Slides

  • A Friendly Introduction to Representation Theory

    • February 2021 | Mathematical and Statistical Sciences Graduate Colloquium (Virtual) | University of Alberta
    • Abstract: Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. When that abstract algebraic object is being represented on a finite-dimensional vector space, its elements are described by matrices and its algebraic operations are described by matrix multiplication and matrix addition. The power of such methodology allows one to reduce abstract algebra problems to linear algebra problems. In this talk, we will introduce the basic notions of representation theory with a focus on group representations.

  • Dualizing Algebras

    • January 2020 | Graduate Algebra Seminar | University of Alberta
    • Abstract: An algebra is traditionally defined as a certain object in the category of modules over some ring or field. We shall realize that such definition can be generalized to work in any monoidal category and can moreover be dualized, introducing the concept of a coalgebra.

  • Introduction to Hopf Algebras

    • December 2017 | University of Western Ontario
    • Abstract: By dualizing the definition of an algebra, we obtain a notion of a coalgebra. Combining these notions with some extra compatibility conditions, one arrives at the definition of a Hopf algebra. We shall discuss why Hopf algebras are important and provide some examples.

  • Yetter-Drinfel’d Modules and the Radford Projection Theorem

    • April 2017 | Bachelor Thesis Public Presentation | Memorial University of Newfoundland
    • Abstract: In the theory of groups, the semidirect product is an approach to decompose a group into a product of two subgroups with one being normal. We shall examine a Hopf algebra analogue of the semidirect product, called the Radford biproduct.

  • Olshansky’s Modification of Golod’s Example of Non-Nilpotent Finitely Generated Nil-Algebras

    • October 2016 | Algebra Seminar | Memorial University of Newfoundland
    • Abstract: Burnside’s problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. A counter-example was provided by Golod in the 1960’s and in this talk we shall provide Olshanky’s version of such counter-example.