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1. A category $$\mathcal{O}$$ for oriented matroids

   https://doi.org/10.4171/JCA/71  

   Kowalenko, Ethan; Mautner, Carl (June 2023, Journal of Combinatorial Algebra)

   We associate to a sufficiently generic oriented matroid program and choice of linear system of parameters a finite-dimensional algebra, whose representation theory is analogous to blocks of Bernstein—Gelfand—Gelfand category\mathcal{O}. When the data above comes from a generic linear program for a hyperplane arrangement, we recover the algebra defined by Braden—Licata—Proudfoot—Webster. Applying our construction to non-linear oriented matroid programs provides a large new class of algebras. For Euclidean oriented matroid programs, the resulting algebras are quasi-hereditary and Koszul, as in the linear setting. In the non-Euclidean case, we obtain algebras that are not quasi-hereditary and not known to be Koszul, but still have a natural class of standard modules and satisfy numerical analogues of quasi-heredity and Koszulity on the level of graded Grothendieck groups. 

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2. Embeddings of von Neumann algebras into uniform Roe algebras and quasi-local algebras

   https://doi.org/10.1016/j.jfa.2023.110186  

   Baudier, Florent P; Braga, Bruno M; Farah, Ilijas; Vignati, Alessandro; Willett, Rufus (January 2024, Journal of Functional Analysis)

   We study which von Neumann algebras can be embedded into uniform Roe algebras and quasi-local algebras associated with a uniformly locally finite metric space X. Under weak assumptions, these C*-algebras contain embedded copies of certain matrix algebras. We aim to show they cannot contain any other von Neumann algebras. One of our main results shows that the only embedded von Neumann algebras are the “obvious” ones. 

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3. Ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras

   https://doi.org/10.1088/1751-8121/ac552c  

   Yan, Bowen; Chen, Penghua; Cui, Shawn X (April 2022, Journal of Physics A: Mathematical and Theoretical)

   Abstract Kitaev’s quantum double model is a family of exactly solvable lattice models that realize two dimensional topological phases of matter. The model was originally based on finite groups, and was later generalized to semi-simple Hopf algebras. We rigorously define and study ribbon operators in the generalized quantum double model. These ribbon operators are important tools to understand quasi-particle excitations. It turns out that there are some subtleties in defining the operators in contrast to what one would naively think of. In particular, one has to distinguish two classes of ribbons which we call locally clockwise and locally counterclockwise ribbons. Moreover, we point out that the issue already exists in the original model based on finite non-abelian groups, but it seems to not have been noticed in the literature. We show how certain common properties would fail even in the original model if we were not to distinguish these two classes of ribbons. Perhaps not surprisingly, under the new definitions ribbon operators satisfy all properties that are expected. For instance, they create quasi-particle excitations only at the end of the ribbon, and the types of the quasi-particles correspond to irreducible representations of the Drinfeld double of the input Hopf algebra. However, the proofs of these properties are much more complicated than those in the case of finite groups. This is partly due to the complications in dealing with general Hopf algebras rather than group algebras. 

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4. DHR bimodules of quasi-local algebras and symmetric quantum cellular automata

   https://doi.org/10.4171/QT/216  

   Jones, Corey (October 2024, Quantum Topology)

   For a net of C*-algebras on a discrete metric space, we introduce a bimodule version of the DHR tensor category and show that it is an invariant of quasi-local algebras under isomorphisms with bounded spread. For abstract spin systems on a latticeL\subseteq \mathbb{R}^{n}satisfying a weak version of Haag duality, we construct a braiding on these categories. Applying the general theory to quasi-local algebrasAof operators on a lattice invariant under a (categorical) symmetry, we obtain a homomorphism from the group of symmetric QCA to\mathbf{Aut}_{\mathrm{br}}(\mathbf{DHR}(A)), containing symmetric finite-depth circuits in the kernel. For a spin chain with fusion categorical symmetry\mathcal{D}, we show that the DHR category of the quasi-local algebra of symmetric operators is equivalent to the Drinfeld center\mathcal{Z}(\mathcal{D}). We use this to show that, for the double spin-flip action\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\curvearrowright \mathbb{C}^{2}\otimes \mathbb{C}^{2}, the group of symmetric QCA modulo symmetric finite-depth circuits in 1D contains a copy ofS_{3}; hence, it is non-abelian, in contrast to the case with no symmetry. 

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5. Planar Algebras in Braided Tensor Categories

   https://doi.org/10.1090/memo/1392  

   Henriques, André; Penneys, David; Tener, James (February 2023, Memoirs of the American Mathematical Society)

   We generalize Jones’ planar algebras by internalising the notion to a pivotal braided tensor category $\mathcal{C}$. To formulate the notion, the planar tangles are now equipped with additional ‘anchor lines’ which connect the inner circles to the outer circle. We call the resulting notion ananchored planar algebra. If we restrict to the case when $\mathcal{C}$is the category of vector spaces, then we recover the usual notion of a planar algebra. Building on our previous work on categorified traces, we prove that there is an equivalence of categories between anchored planar algebras in $\mathcal{C}$and pivotal module tensor categories over $\mathcal{C}$equipped with a chosen self-dual generator. Even in the case of usual planar algebras, the precise formulation of this theorem, as an equivalence of categories, has not appeared in the literature. Using our theorem, we describe many examples of anchored planar algebras. 

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