Quasi-Frobenius algebras in finite tensor categories

Abstract

We introduce the notion of a quasi-Frobenius algebra in a finite tensor category $\mathcal{C}$ and give equivalent conditions for an algebra in $\mathcal{C}$ to be quasi-Frobenius. A quasi-Frobenius algebra in $\mathcal{C}$ is not necessarily Frobenius, however, we show that an algebra $A$ in $\mathcal{C}$ is quasi-Frobenius if and only if $A$ is Morita equivalent to a Frobenius algebra in $\mathcal{C}$. We also show that the class of symmetric Frobenius algebras in $\mathcal{C}$ is closed under the Morita equivalence provided that $\mathcal{C}$ is pivotal so that the symmetricity makes sense.

Figures (3)

• Figure 1: Proof of Lemma \ref{['lem:internal-Yoneda-1']}
• Figure 2: Proof of Lemma \ref{['lem:rel-Serre-trace-2']}
• Figure 3: Verification that $\theta$ is a morphism of $\mathcal{C}$-module functors

Theorems & Definitions (108)

• Definition 1.1: $=$ Definition \ref{['def:QF-algebras']}
• Theorem 1.2: $=$ Theorem \ref{['thm:QF-algebras']}
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5