Division algebras in monoidal categories
Jacob Kesten, Chelsea Walton
TL;DR
The paper advances the theory of division algebras from classical field-based algebra to monoidal categories by introducing module-theoretic notions (simplistic, essential) and a monad-theoretic notion (monadic division algebras). It establishes precise relationships among these notions, showing, for example, that in abelian, rigid settings essential division algebras coincide with monadic ones, and that in pivotal multifusion categories left and right division notions agree. The work extends these definitions to multifusion and pivotal multifusion categories via internal End algebras and Ostrik’s Morita-equivalence framework, and provides concrete examples, including End algebras from simple objects and monad-induced constructions on $\mathsf{Set}$ and $\mathsf{Vec}$. By connecting division algebras to module category structure and monads, the paper broadens Morita theory in monoidal categories and opens avenues for Frobenius-type structural results and classification in finite monoidal settings.
Abstract
This work adapts the equivalent definitions of division algebras over a field into multiple types of division algebras in a monoidal category. Examples and consequences of these definitions are then established in various monoidal settings.