2025/01/28 by David Jaklitsch, Jaklitsch, David, Harshit Yadav +1 · 1 citation
Mathematics · #Rings, Modules, and Algebras #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications
paper · pdf · doi:10.48550/arxiv.2501.16978
We call a tensor functor F:C\toD between finite tensor categories ⊗-Frobenius if its left and right adjoints are isomorphic as C-bimodule functors. We give several characterizations of this notion -- most notably, F is ⊗-Frobenius if and only if the centralizer Z(F D F) is unimodular. We use them to analyze how actions on module categories behave under pullback along F. For perfect functors, we show that twisting a D-module category M along F preserves exactness, and that pivotality, unimodularity, and sphericality are preserved whenever F is ⊗-Frobenius (or, more generally, Frobenius with respect to M). Applications include: (i) explicit criteria for ⊗-Frobenius functors arising from bialgebra maps f : H' → H between finite-dimensional Hopf algebras; and (ii) criteria ensuring that objects of internal natural transformations are (symmetric) Frobenius algebras in Z(C). Along the way we show that central tensor functors are Frobenius iff they are ⊗-Frobenius and that any tensor functor between separable fusion categories is ⊗-Frobenius, answering questions of Flake-Laugwitz-Posur.