Hopf actions of Bosonizations on path algebras of quivers

Symmetry is a core concept in numerous areas of mathematics, manifesting itself not only in visual forms such as geometric shapes but also in more abstract structures such as algebraic and topological systems. Symmetry fundamentally refers to an object’s invariance under certain transformations, such as rotations, reflections, or translations, and it plays a crucial role in both pure and applied mathematics. In modern mathematics, symmetries are rigorously encoded by group actions, a concept from group theory that formalizes how a group, which consists of a set of elements combined with an operation, acts on a mathematical object.

Beyond classical symmetries, modern mathematics explores generalized symmetry structures that extend traditional group-based frameworks. This shift is particularly evident in mathematical physics, where certain problems require more flexible algebraic structures to describe symmetries in quantum systems. Hopf algebras emerge as a natural extension, providing a richer framework that accommodates deformations of classical symmetries, leading to the development of quantum groups.

The development of quantum groups traces back to the 1980s, by the work of Vladimir Drinfeld and Michio Jimbo. They both independently formulated these structures while studying solutions to the Yang-Baxter equation, a key equation in statistical mechanics and quantum field theory. The connection between quantum groups and Hopf algebras is fundamental. Hopf algebras provide the algebraic framework to describe both the algebraic operations and the additional structures of an antipode and counit that quantum groups possess.

Building on these ideas, my research explores the interplay between Hopf algebras and quantum symmetries. Specifically, I study the Bosonizations of the quantum linear spaces with group algebras, a construction that results in a Hopf algebra capturing both classical and quantum symmetries. Additionally, I am classifying Hopf actions on path algebras of quivers, investigating how these algebraic structures encode symmetry in a noncommutative setting. By analyzing these actions, my work contributes to a deeper understanding of how Hopf algebras interact with fundamental algebraic and combinatorial structures, with potential applications in representation theory and mathematical physics.