Algebraic Quantum Symmetry

Symmetry is one of the oldest notions in mathematics. Many algebraic structures have been introduced to axiomatize this notion, starting with groups in the mid-19th century. Arguably, in each area, the mathematical tools that capture symmetry have an underlying structure that is algebraic; structures that are known as a bialgebra, a Hopf algebra, or a Hopf-type algebra.

Nowadays, connections between these settings are examined through the lens of category theory that allows for the use of special structures known as monoidal categories, which have numerous applications including quantum information, quantum field theory and string theory. The main goal of the project is to study symmetries of algebra objects within monoidal categories, especially co/representation categories of Hopf-type algebras.

This project will fund undergraduate research and the PI will continue their advocacy work for members of underrepresented groups in the mathematical sciences.

Given an object X, a symmetry of X is a property-preserving transformation from X to itself, and the collection of invertible symmetries of X forms a group: the automorphism group of X. Since then, generalizations of groups have been introduced to capture the symmetries of not only objects, but also of function algebras of objects that cannot be observed (e.g., objects in quantum physics).

This move from “classical symmetry” to “quantum symmetry” has its origins in quantum mechanics, and arises in active research areas such as conformal field theory, low-dimensional topology, and operator algebras. In examining the various settings of symmetry of algebras beyond the framework of "classical symmetry", comprised of groups actions on commutative algebras, the PI will continue their work in "quantum symmetry" involving co/actions of bialgebras, or of Hopf algebras, on noncommutative algebras with a trivial base.

The PI will also delve further into "weak quantum symmetry" involving co/actions of weak bialgebras, or of weak Hopf algebras, on noncommutative algebras with a non-trivial base. Moreover, the PI will examine algebras via "categorical quantum symmetry”: This pertains to studying algebras in general monoidal categories, not necessarily in co/representation categories of (weak) bi/Hopf algebras, including several types of semisimple monoidal categories, for example, fusion categories, and modular tensor categories, both semisimple and nonsemisimple.

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.