This paper explains the methods and algorithms we used to obtain explicit F-symbols, R-symbols, and pivotal coefficients of what we believe are all multiplicity-free pivotal complex fusion categories up to rank 7. Particular attention is given to how the individual steps for solving the consistency equations are done with Anyonica, a software package we developed for working with fusion categories. Gauge and automorphism equivalence are reviewed and a method to remove equivalent categories via a gauge-split basis is given. A census of all multiplicity-free fusion categories up to rank 7, along with all compatible braided and pivotal structures, is presented, and comments on their general properties are given.
Abstract Given a $$\hbox {II}_1$$ II 1 factor M , a $$\hbox {W}^*$$ W ∗ -subalgebra $$Q\subset M$$ Q ⊂ M is compressible if for any $$\varepsilon >0$$ ε > 0 there exists a finite set of unitary elements $${\mathcal {U}}_0\subset {\mathcal {U}}(M)=\mathcal U(M\otimes 1)$$ U 0 ⊂ U ( M ) = U ( M ⊗ 1 ) such that $$\Vert \frac{1}{|{\mathcal {U}}_0|}\sum _{u\in \mathcal U_0} uxu^* -E_{1\otimes {\mathbb {M}}_K({\mathbb {C}})}(x)\Vert \le \varepsilon $$ ‖ 1 | U 0 | ∑ u ∈ U 0 u x u ∗ - E 1 ⊗ M K ( C ) ( x ) ‖ ≤ ε , $$\forall K\ge 1$$ ∀ K ≥ 1 , $$\forall x\in (Q\otimes {\mathbb {M}}_K({\mathbb {C}}))_1$$ ∀ x ∈ ( Q ⊗ M K ( C ) ) 1 . Any $$\hbox {W}^*$$ W ∗ -subalgebra Q in a $$\hbox {II}_1$$ II 1 factor M which admits a diffuse $$\hbox {W}^*$$ W ∗ -algebra $$Q_0\subset M$$ Q 0 ⊂ M that’s free independent to Q , is compressible in M . We prove that if $$Q\subset M$$ Q ⊂ M is compressible, then $$_NL^2M_Q$$ N L 2 M Q contains a copy of the coarse $$N-Q$$ N - Q bimodule for any AFD subalgebra $$N\subset M$$ N ⊂ M . We use this result to provide examples of inclusions of factors $$M\subset {\mathcal {M}}$$ M ⊂ M that are ergodic but not AFD-ergodic, even after stabilizing by $$\mathcal B(\ell ^2{\mathbb {N}})$$ B ( ℓ 2 N ) .
We establish several geometric characterizations and rigidity results for 3-dimensional asymptotically locally hyperbolic (ALH) static spaces with horizon boundary. Notably, a 3-dimensional ALH static space with toroidal infinity and strictly non-spherical horizons is isometric to a toroidal Kottler metric. Furthermore, we show that the surface gravity of a static horizon with spherical infinity is bounded below by √(3) , with equality achieved only by the critical AdS-Schwarzschild metric. Consequently, Poincaré-Einstein fillings of S^2× S^1(λ ) arising from these spaces have length parameter λ≤ (√(3))^-1 , which supports a recent conjecture of Chang-Yang-Zhang[18]. Finally, static horizons with hyperbolic infinity and non-negative Chruściel-Herzlich mass obey the reverse Riemannian Penrose inequality. In conjunction with work of Ge-Wang-Wu-Xia [29], we use this fact to obtain uniqueness of static ALH graphs with hyperbolic infinities. These results follow from a generalization of the Minkowski inequality in AdS-Schwarzschild space due to Brendle-Hung-Wang [15]. Using optimal coefficients for the sub-static Heintze-Karcher inequality from [24], we construct a new monotone quantity under inverse mean curvature flow (IMCF) in static spaces with negative cosmological constant. Another fundamental tool developed in this paper is a regularity theorem for IMCF in ALH manifolds. Specifically, we prove that a weak solution of IMCF in an ALH 3-manifold with horizon boundary is eventually smooth. This extends the regularity theorem for a spherical infinity due to Shi-Zhu [52].
We study the conditional measures of every invariant probability measure for every C^1+γ diffeomorphism of a closed Riemannian manifold, and we show that they always admit an asymptotic local product structure. This is a direct application of the notion of tubular dimension which we introduce and study in the manuscript. As an additional application we prove a bound on any two consecutive conditional entropies in terms of volume growth.
