Indecomposable Finite Semisimple Bimodule Category

• Indecomposable finite semisimple bimodule categories are algebraic structures that generalize module categories, featuring pivotal bimodule actions and ensuring no nontrivial decompositions.
• They enable advanced techniques in representation theory and TQFT through character theory, induction–restriction formalism, and the study of categorical centers.
• Their realization via fusion categories, Morita equivalence, and pivotal functors underpins applications such as constructing 4-manifold invariants and classifying subfactors.

An indecomposable finite semisimple bimodule category is a central object in the categorical formulation of higher representation theory, topological quantum field theory, and subfactor theory. Such a category generalizes the notion of module categories over multifusion (or fusion) categories, extending it to the setting of bimodule actions, Morita theory, and categorical duality. The indecomposability condition ensures structural simplicity and the absence of nontrivial decompositions, allowing powerful character-theoretic and homological techniques. These categories feature prominently in the paper of categorical centers, induction–restriction formalism, realization of bimodules as standard invariants of subfactors, classification of module categories, and in the diagrammatic calculus for manifold invariants.

1. Multifusion Categories and Bimodule Categories

A multifusion category $\mathcal{C}$ over an algebraically closed field of characteristic 0 is a $k$-linear, semisimple, rigid monoidal abelian category with finitely many simple objects (up to isomorphism), finite-dimensional hom spaces, and no nontrivial direct sum decompositions as tensor categories. If the unit object is simple, $\mathcal{C}$ is a fusion category (Deshpande, 2016).

A $\mathcal{C}$-bimodule category $\mathcal{M}$ is a $k$-linear semisimple abelian category with commuting left and right exact actions of $\mathcal{C}$, satisfying associativity and unit constraints. Finiteness refers to having finitely many simple objects (up to isomorphism) and finite-dimensional hom spaces. The indecomposability condition means $\mathcal{M}$ cannot be written as a nontrivial direct sum of (bi)module subcategories. Equivalently, $\mathcal{M}$ is indecomposable as a $\mathcal{C}$-bimodule if and only if its endomorphism category $$\operatorname{End}_{(\mathcal{C},\mathcal{D})}(\mathcal{M})$$ is a fusion category for bimodules over $(\mathcal{C},\mathcal{D})$ (Meusburger et al., 24 Nov 2025).

2. Invertibility, Morita Theory, and Brauer–Picard Group

An invertible $\mathcal{C}$-bimodule category $\mathcal{M}$ admits a two-sided inverse $\mathcal{M}^{\mathrm{op}}$ under the relative Deligne tensor product: $$\mathcal{M} \boxtimes_{\mathcal{C}} \mathcal{M}^{\mathrm{op}} \simeq \mathcal{C}$$. The set of equivalence classes of invertible finite semisimple (multi)fusion $\mathcal{C}$-bimodules forms the Brauer–Picard group $\operatorname{BrPic}(\mathcal{C})$. By [ENO2, Thm. 1.1], the Drinfeld center functor $$Z(-)\colon \operatorname{BrPic} \to \operatorname{EqBr}$$ is fully faithful, implying invertible bimodule categories correspond to braided equivalences of centers. Any indecomposable bimodule $\mathcal{M}$ is Morita equivalent to the trivial $\mathcal{C}$-bimodule via $\mathcal{M}$ itself (Deshpande, 2016, Femic et al., 2014).

In the case where $\mathcal{C} = \operatorname{Rep}(H)$ for a finite-dimensional Hopf algebra $H$, invertible exact $\operatorname{Rep}(H)$-bimodule categories are classified by $H$-biGalois objects, with the group $\operatorname{BiGal}(H)$ embedding into $\operatorname{BrPic}(\operatorname{Rep}(H))$ (Femic et al., 2014).

3. Centers, Induction–Restriction, and Grothendieck Theory

Given an indecomposable (multi)fusion category $\mathcal{C}$ and an invertible $\mathcal{C}$-bimodule $\mathcal{M}$, the (relative Drinfeld) center $Z_\mathcal{C}(\mathcal{M})$ is defined as the category of $\mathcal{C}$-bimodule functors $$\operatorname{Fun}_{\mathcal{C}|\mathcal{C}}(\mathcal{C}, \mathcal{M})$$. Each object is a pair $(m, B_{-})$, where $m \in \mathcal{M}$ and $B_c: c \otimes m \cong m \otimes c$ isomorphisms, satisfying hexagon compatibility. $Z_\mathcal{C}(\mathcal{M})$ is an invertible left module over the braided fusion category $Z(\mathcal{C})$ (Deshpande, 2016).

There exist pivotal functors:

• Restriction/Forgetful: $$\zeta_\mathcal{M}: Z_\mathcal{C}(\mathcal{M}) \to \mathcal{M}$$, forgetting the half-braiding;
• Induction: $$\chi_\mathcal{M}: \mathcal{M} \to Z_\mathcal{C}(\mathcal{M})$$, its right adjoint.

These functors generalize Lusztig’s restriction and induction for character sheaves, and their composition is central to understanding the relationship between bimodule category theory and the character tables of Grothendieck rings. In particular, the decomposition formulas express the multiplicities $$m_{A,m} = [A: \chi_\mathcal{M}(m)] = [m: \zeta_\mathcal{M}(A)]$$ in terms of irreducible characters $E$ of the Grothendieck algebra and the central map $$S: K_{Q_{ab}}(Z(\mathcal{C})) \hookrightarrow Z(K_{Q_{ab}}(\mathcal{C}))$$ (Deshpande, 2016).

4. Concrete Realizations and Classification

Every finite semisimple indecomposable C*-tensor category $C$ arises as the full bimodule category $\operatorname{Bimod}(M)$ of some $\mathrm{II}_1$ factor $M$, with a fully faithful tensor functor equivalence. The explicit construction uses subfactor planar algebras associated to $C$, Popa’s reconstruction theorem, and amalgamated free product techniques to ensure all finite-index bimodules are obtained. In this correspondence, simple objects of $C$ biject with irreducible finite-index bimodules of $M$, and their tensor products correspond to Connes’ relative tensor product structure (Falguières et al., 2011).

In the special case of finite-dimensional Hopf algebras $H$, the classification of indecomposable finite semisimple exact $\operatorname{Rep}(H)$-bimodule categories is explicitly worked out for $T_q$ (the Taft algebra): any such bimodule is equivalent to a module category over a coideal subalgebra, falling into precisely five families, with biGalois objects corresponding to invertible bimodules and forming an explicit subgroup in the Brauer–Picard group (Femic et al., 2014).

5. Spherical Structures, Traces, and Pivotality

If $\mathcal{C}$ is spherical and $\mathcal{M}$ is an indecomposable bimodule admitting a compatible bimodule trace, then $Z(\mathcal{C})$ becomes a modular tensor category, and $Z_\mathcal{C}(\mathcal{M})$ acquires a module trace. A trace on $\mathcal{M}$ is a family of linear maps $$\operatorname{tr}_m: \operatorname{End}_\mathcal{M}(m) \to \mathbb{C}$$, cyclic and nondegenerate, compatible with both module actions.

The uniqueness (up to scalar) of such a trace on indecomposable $\mathcal{M}$ ensures pivotality of the dual category $$\operatorname{End}_{(\mathcal{C},\mathcal{D})}(\mathcal{M})$$. In the presence of spherical structures, the multiplicity formulas for simple objects in the image of induction/restriction functors simplify into crossed $S$-matrix formulas, generalizing Verlinde-type formulae and linking to modular data (Deshpande, 2016, Meusburger et al., 24 Nov 2025).

6. Applications: Diagrammatic Calculus and 4-Manifold Invariants

Indecomposable finite semisimple bimodule categories equipped with a bimodule trace are crucial in the construction of new 4-manifold invariants via trisection diagrams (Meusburger et al., 24 Nov 2025). The algebraic data for such invariants consist of spherical fusion categories $\mathcal{A}$, $\mathcal{C}$, $\mathcal{D}$, an indecomposable finite semisimple $(\mathcal{C},\mathcal{D})$-bimodule category $\mathcal{M}$ with bimodule trace, and a pivotal functor $$\Phi: \mathcal{A} \to \operatorname{End}_{(\mathcal{C},\mathcal{D})}(\mathcal{M})$$. The interaction of these data in a diagrammatic (surface) calculus incorporates fusion, the bimodule action, dualities, and the pivotal structure. The indispensable property is that $\operatorname{End}(\mathcal{M})$ is fusion (enabling duals and integrals) and pivotal (from the trace), ensuring the resulting invariants are well-defined and topologically meaningful.

7. Examples and Further Developments

Typical examples include:

• Graded vector spaces with $G$-action and associator twist, leading to module categories classified by $G$-sets and group cohomology classes;
• Representation categories of semisimple Hopf algebras, with module (or bimodule) categories arising from (bi)Galois objects or fibre functors;
• Modular fusion categories as bimodule categories over themselves, with the standard trace.

The full landscape of indecomposable finite semisimple bimodule categories includes deep connections with categorical Morita theory, the structure and classification of factor bimodules, and the explicit realization of fusion data in both quantum algebra and low-dimensional topology (Deshpande, 2016, Falguières et al., 2011, Femic et al., 2014, Meusburger et al., 24 Nov 2025).