The fusion algebra of bimodule categories

TL;DR

The paper establishes a complete algebraic bridge between the complexified Grothendieck ring $\mathcal{F}=K_0(\mathcal{C}_{A|A})\otimes\mathbb{C}$ of a bimodule category and the endomorphism algebra $\mathcal{E}=\bigoplus_{i,j\in I}\mathrm{End}_{\mathbb{C}}(\mathrm{Hom}_{A|A}(\alpha^{+}(U_i),\alpha^{-}(U_j)))$, via an explicit isomorphism $\Phi$ built from bimodule endomorphisms $D_X^{UV}$. It proves $\Phi$ is a unital algebra isomorphism (Lemmas 2 and 3), yielding a block-diagonalization with blocks of size $n_{(i,j)}=z_{i,j}=\dim_{\mathbb{C}}\mathrm{Hom}_{\mathcal{C}_{\mathcal{M}}^{*}}(\alpha^{+}(U_i),\alpha^{-}(U_j))$, and giving a formula for the fusion constants $N^{\kappa''}_{\kappa\kappa'}$ in terms of the basis-change matrix $d$ with entries $d_{\kappa}^{ij;\alpha\beta}$. The approach identifies $z(A)_{i,j}=\dim_{\mathbb{C}}\mathrm{Hom}_{A|A}(\alpha^{+}_{A}(U_i),\alpha^{-}_{A}(U_j))$ with the block sizes, providing a concrete, endomorphism-based description of $\mathcal{F}$ and relating the invariant data to the tensor-unit endomorphisms in $\mathcal{C}$.

Abstract

We establish an algebra-isomorphism between the complexified Grothendieck ring F of certain bimodule categories over a modular tensor category and the endomorphism algebra of appropriate morphism spaces of those bimodule categories. This provides a purely categorical proof of a conjecture by Ostrik concerning the structure of F. As a by-product we obtain a concrete expression for the structure constants of the Grothendieck ring of the bimodule category in terms of endomorphisms of the tensor unit of the underlying modular tensor category.