Quivers in quantum symmetry: a path algebra framework for algebras in tensor categories

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The concept of "symmetry" has been recognized since time immemorial, because of its ubiquitous presence in the natural world. Symmetries can be immediately visually apparent, as with mirror images or circles, but also may be more abstract. For example, solutions to an equation may have symmetries that help us better understand them, or molecules may have symmetries that affect how they interact with one another. There are a number of ways to use the language of mathematics to formalize the concept of symmetry, and some of these lie within the field of Abstract Algebra. Informally, this project will work with symmetries of an object computationally, somewhat like familiar number systems used in everyday life, but more complicated. The project lies within the newly evolving field of "quantum symmetry" which allows for greater flexibility in the concept, at the expense of losing some intuition. However, it can be formalized just as rigorously using mathematics, specifically in the language of Hopf algebras and tensor categories. The proposed work broadens participation in mathematics by having specific subprojects for a diverse group of doctoral students, which will prepare them for their own research careers, in academics, business, industry, or government.More precisely, this project extends the foundations of representation theory of finite dimensional associative algebras to the setting of algebras in tensor categories. The PI will particularly focus on finite tensor categories and expects the strongest results for fusion categories. The proposed extension of quivers and their representations to the setting of finite tensor categories will accommodate richer underlying structures than just vector spaces, such as the action of a group by automorphisms or a Lie algebras by derivations, as well as vector spaces graded by groups, and even non-classical structures where the objects have fractional (Frobenius-Perron) dimension, and thus cannot be considered to have an underlying vector space in any natural way. A specific class of tensor categories which will receive detailed attention are representation categories of Hopf algebras. This includes many examples of interest across broad areas of mathematics, such as quantum groups. This project is jointly funded by the Algebra and Number Theory Program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.