2021/06/03 by Chelsea Walton, Walton, Chelsea, Harshit Yadav +1
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2106.01999
openalex publication_date 2021/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop filtered-graded techniques for algebras in monoidal categories with the main goal of establishing a categorical version of Bongale's 1967 result: A filtered deformation of a Frobenius algebra over a field is Frobenius as well. Towards the goal, we first construct a monoidal associated graded functor, building on prior works of Ardizzoni-Menini, of Galatius et al., and of Gwillian-Pavlov. Next, we produce equivalent conditions for an algebra in a rigid monoidal category to be Frobenius in terms of the existence of categorical Frobenius form; this builds on work of Fuchs-Stigner. These two results of independent interest are then used to achieve our goal. As an application of our main result, we show that any exact module category over a symmetric finite tensor category C is represented by a Frobenius algebra in C. Several directions for further investigation are also proposed.