A Frobenius algebra object $A$ in a tensor category $\mathcal C$ is said to be connected if $\text{Hom}_{\mathcal C}(\mathbb{1}, A)$ is a one dimensional vector space, where $\mathbb {1} $ denotes the tensor unit of $\mathcal C$.
Can someone provide examples of connected Frobenius algebras that are non-semisimple as an object of a tensor category?
In [FSS12, Proposition 2.10], it is demonstrated that for a finite-dimensional ribbon factorizable (so unimodular) Hopf algebra $ H $, the dual $ H^* $ possesses a Frobenius algebra structure within the category $\mathrm{Bimod}(H)$, although it has a different comultiplication and counit.
According to [EGNO15, Corollary 7.20.4], if $\mathcal{C}$ is a unimodular tensor category, then $\underline{\mathrm{Hom}}(\mathbf{1},\mathbf{1})$ in $\mathcal{C} \boxtimes \mathcal{C}^{\mathrm{op}}$ is identified as its canonical Frobenius algebra and is connected, as demonstrated below. This framework seems to align with [FSS12, Proposition 2.10] when $\mathcal{C} = \mathrm{Rep}(H)$, where $ H $ is a finite-dimensional unimodular Hopf algebra. As detailed in [EGNO15, Example 7.20.6], $ H^* $ serves as the canonical Frobenius algebra in the category $\mathrm{Rep}(H \otimes H^{\mathrm{cop}})$, albeit featuring a distinct comultiplication and counit.
In [LW23, Section 6], various rigid Frobenius algebras within non-semisimple modular tensor categories are explored. Recall from [LW23, Definition 3.8] that an algebra $ A $ in $\mathcal{C}$ is termed rigid Frobenius if it is connected, commutative, and special Frobenius.
Canonical Frobenius algebra and connectedness
In [EGNO15, Corollary 7.20.4], the Frobenius algebra $\underline{\mathrm{Hom}}(\mathbf{1},\mathbf{1})$ in $\mathcal{C} \boxtimes \mathcal{C}^{\mathrm{op}}$ is discussed for a unimodular multitensor category $\mathcal{C}$. However, as noted with Mainak, we have the following:
Theorem: The Frobenius algebra $\underline{\mathrm{Hom}}(\mathbf{1},\mathbf{1})$ is connected if and only if $\mathcal{C}$ is tensor.
Proof: Consider a multitensor category $\mathcal{D}$ and a $\mathcal{D}$-module category $\mathcal{M}$, with objects $M_1$ and $M_2$ in $\mathcal{M}$. According to [EGNO15, (7.20)], $\underline{\mathrm{Hom}}(M_1, M_2)$ is defined via the natural isomorphism: $$ \mathrm{Hom}_{\mathcal{M}}(X \otimes M_1, M_2) \simeq \mathrm{Hom}_{\mathcal{D}}(X, \underline{\mathrm{Hom}}(M_1, M_2)). $$
Assume $\mathcal{M}$ is also monoidal with unit $\mathbf{1}_{\mathcal{M}}$, and let $\mathbf{1}_{\mathcal{D}}$ be the unit of $\mathcal{D}$. Set $M_1 = M_2 = \mathbf{1}_{\mathcal{M}}$ and $X = \mathbf{1}_{\mathcal{D}}$, yielding: $$ \mathrm{Hom}_{\mathcal{M}}(\mathbf{1}_{\mathcal{M}}, \mathbf{1}_{\mathcal{M}}) \simeq \mathrm{Hom}_{\mathcal{D}}(\mathbf{1}_{\mathcal{D}}, \underline{\mathrm{Hom}}(\mathbf{1}_{\mathcal{M}}, \mathbf{1}_{\mathcal{M}})). $$
Thus, $\underline{\mathrm{Hom}}(\mathbf{1}_{\mathcal{M}}, \mathbf{1}_{\mathcal{M}})$ is connected if and only if $\mathbf{1}_{\mathcal{M}}$ is linear-simple, meaning $\mathrm{Hom}_{\mathcal{M}}(\mathbf{1}_{\mathcal{M}}, \mathbf{1}_{\mathcal{M}})$ is one-dimensional. Therefore, for a multitensor category $\mathcal{M}$, $\underline{\mathrm{Hom}}(\mathbf{1}_{\mathcal{M}}, \mathbf{1}_{\mathcal{M}})$ is connected if and only if $\mathcal{M}$ is a tensor category. The theorem follows by setting $\mathcal{D} = \mathcal{C} \boxtimes \mathcal{C}^{\mathrm{op}}$ and $\mathcal{M} = \mathcal{C}$. $\square$
References
[EGNO15] Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor. Tensor Categories. Mathematical Surveys and Monographs, 205. American Mathematical Society, Providence, RI, 2015. xvi+343 pp.
[FSS12] Fuchs, Jürgen; Schweigert, Christoph; Stigner, Carl. Modular invariant Frobenius algebras from ribbon Hopf algebra automorphisms. J. Algebra 363 (2012), 29--72.
[LW23] Laugwitz, Robert; Walton, Chelsea. Constructing non-semisimple modular categories with local modules. Comm. Math. Phys. 403 (2023), no. 3, 1363--1409.
