Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDistortion for multifactor bimodules and representations of multifusion categories
74
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We call a von Neumann algebra with finite dimensional center a multifactor. We introduce an invariant of bimodules over $rm II_1$ multifactors that we call modular distortion, and use it to formulate two classification results. We first classify finite depth finite index connected hyperfinite $rm II_1$ multifactor inclusions $Asubset B$ in terms of the standard invariant (a unitary planar algebra), together with the restriction to $A$ of the unique Markov trace on $B$. The latter determines the modular distortion of the associated bimodule. Three crucial ingredients are Popas uniqueness theorem for such inclusions which are also homogeneous, for which the standard invariant is a complete invariant, a generalized version of the Ocneanu Compactness Theorem, and the notion of Morita equivalence for inclusions. Second, we classify fully faithful representations of unitary multifusion categories into bimodules over hyperfinite $rm II_1$ multifactors in terms of the modular distortion. Every possible distortion arises from a representation, and we characterize the proper subset of distortions that arise from connected $rm II_1$ multifactor inclusions.
rate research
Read More
A bicommutant category is a higher categorical analog of a von Neumann algebra. We study the bicommutant categories which arise as the commutant $mathcal{C}$ of a fully faithful representation $mathcal{C}tooperatorname{Bim}(R)$ of a unitary fusion category $mathcal{C}$. Using results of Izumi, Popa, and Tomatsu about existence and uniqueness of representations of unitary (multi)fusion categories, we prove that if $mathcal{C}$ and $mathcal{D}$ are Morita equivalent unitary fusion categories, then their commutant categories $mathcal{C}$ and $mathcal{D}$ are equivalent as bicommutant categories. In particular, they are equivalent as tensor categories: [ Big(,,mathcal{C},,simeq_{text{Morita}},,mathcal{D},,Big) qquadLongrightarrowqquad Big(,,mathcal{C},,simeq_{text{tensor}},,mathcal{D},,Big). ] This categorifies the well-known result according to which the commutants (in some representations) of Morita equivalent finite dimensional $rm C^*$-algebras are isomorphic von Neumann algebras, provided the representations are `big enough. We also introduce a notion of positivity for bi-involutive tensor categories. For dagger categories, positivity is a property (the property of being a $rm C^*$-category). But for bi-involutive tensor categories, positivity is extra structure. We show that unitary fusion categories and $operatorname{Bim}(R)$ admit distinguished positive structures, and that fully faithful representations $mathcal{C}tooperatorname{Bim}(R)$ automatically respect these positive structures.
In this paper we show that Soergel bimodules for finite Coxeter types have only finitely many equivalence classes of simple transitive $2$-representations and we complete their classification in all types but $H_{3}$ and $H_{4}$.
We introduce a notion of bimodule in the setting of enriched $infty$-categories, and use this to construct a double $infty$-category of enriched $infty$-categories where the two kinds of 1-morphisms are functors and bimodules. We then consider a natural definition of natural transformations in this context, and show that in the underlying $(infty,2)$-category of enriched $infty$-categories with functors as 1-morphisms the 2-morphisms are given by natural transformations.
After recalling in detail some basic definitions on Hilbert C*-bimodules, Morita equivalence and imprimitivity, we discuss a spectral reconstruction theorem for imprimitivity Hilbert C*-bimodules over commutative unital C*-algebras and consider some of its applications in the theory of commutative full C*-categories.
The quantum Satake correspondence relates dihedral Soergel bimodules to the semisimple quotient of the quantum $mathfrak{sl}_2$ representation category. It also establishes a precise relation between the simple transitive $2$-representations of both monoidal categories, which are indexed by bicolored $mathsf{ADE}$ Dynkin diagrams. Using the quantum Satake correspondence between affine $mathsf{A}_{2}$ Soergel bimodules and the semisimple quotient of the quantum $mathfrak{sl}_3$ representation category, we introduce trihedral Hecke algebras and Soergel bimodules, generalizing dihedral Hecke algebras and Soergel bimodules. These have their own Kazhdan-Lusztig combinatorics, simple transitive $2$-representations corresponding to tricolored generalized $mathsf{ADE}$ Dynkin diagrams.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
