On Frobenius algebras in rigid monoidal categories
Jurgen Fuchs, Carl Stigner
TL;DR
The paper generalizes Frobenius algebra characterizations from vector spaces to rigid monoidal categories, proving that the three classical notions—$(\Delta,\varepsilon)$-Frobenius, $\kappa$-Frobenius, and $\Phi_{\rho}$-Frobenius—are equivalent in any rigid monoidal category. It then extends to symmetric Frobenius algebras by requiring sovereign structure, showing that symmetric $(\Delta,\varepsilon)$-Frobenius, symmetric $\kappa$-Frobenius, and symmetric $\Phi_{\rho}$-Frobenius are equivalent, with symmetry characterized by $\mho_\varepsilon = id_A$ or $\Phi_{\kappa,\mathrm{l}}=\Phi_{\kappa,\mathrm{r}}$. Nakayama automorphisms are defined and shown to be algebra morphisms, with any two Nakayama automorphisms differing by an inner automorphism; a Frobenius algebra is symmetric iff all Nakayama automorphisms are inner. The work unifies categorical perspectives on Frobenius structures, clarifies their relations under dualities, and provides a robust framework applicable to tensor categories and their applications in quantum algebra and conformal field theory.
Abstract
We show that the equivalence between several possible characterizations of Frobenius algebras, and of symmetric Frobenius algebras, carries over from the category of vector spaces to more general monoidal categories. For Frobenius algebras, the appropriate setting is the one of rigid monoidal categories, and for symmetric Frobenius algebras it is the one of sovereign monoidal categories. We also discuss some properties of Nakayama automorphisms.