  Given a unitary fusion category, one can define the Hilbert space of a
so-called ``anyonic spin-chain'' and nearest neighbor Hamiltonians providing a
real-time evolution. There is considerable evidence that suitable scaling
limits of such systems can lead to $1+1$-dimensional conformal field theories
(CFTs), and in fact, can be used potentially to construct novel classes of
CFTs. Besides the Hamiltonians and their densities, the spin chain is known to
carry an algebra of symmetry operators commuting with the Hamiltonian, and
these operators have an interesting representation as matrix-product-operators
(MPOs). On the other hand, fusion categories are well-known to arise from a von
Neumann algebra-subfactor pair. In this work, we investigate some interesting
consequences of such structures for the corresponding anyonic spin-chain model.
One of our main results is the construction of a novel algebra of MPOs acting
on a bi-partite anyonic chain. We show that this algebra is precisely
isomorphic to the defect algebra of $1+1$ CFTs as constructed by Fr\" ohlich et
al. and Bischoff et al., even though the model is defined on a finite lattice.
We thus conjecture that its central projections are associated with the
irreducible vertical (transparent) defects in the scaling limit of the model.
Our results partly rely on the observation that MPOs are closely related to the
so-called ``double triangle algebra'' arising in subfactor theory. In our
subsequent constructions, we use insights into the structure of the double
triangle algebra by B\" ockenhauer et al. based on the braided structure of the
categories and on $\alpha$-induction. The introductory section of this paper to
subfactors and fusion categories has the character of a review.
