International Journal of Mathematics

Kazhdan and Wenzl classified all rigid tensor categories with fusion ring isomorphic to the fusion ring of the group SU(d). In this paper we consider the C*-analogue of this problem. Given a rigid C*-tensor category 𝒞 with fusion ring isomorphic to the fusion ring of the group SU(d), we can extract a constant q from 𝒞 such that there exists a *-representation of the Hecke algebra H_n(q) into 𝒞. The categorical trace on 𝒞 induces a Markov trace on H_n(q). Using this Markov trace and a representation of H_n(q) in we show that 𝒞 is equivalent to a twist of the category . Furthermore a sufficient condition on a C*-tensor category 𝒞 is given for existence of an embedding of a twist of in 𝒞.

For a given minimal Legendrian submanifold L of a Sasaki–Einstein manifold we construct two families of eigenfunctions of the Laplacian of L and we give a lower bound for the dimension of the corresponding eigenspace. Moreover, in the case the lower bound is attained, we prove that L is totally geodesic and a rigidity result about the ambient manifold. This is a generalization of a result for the standard Sasakian sphere done by Lê and Wang.

In this note, we provide some general discussion on the two main versions in the study of Kähler–Ricci flows over closed manifolds, aiming at smooth convergence to the corresponding Kähler–Einstein metrics with assumptions on the volume form and Ricci curvature form along the flow.

In this article, we study the topology of the family of real analytic germs F : (ℂ³, 0) → (ℂ, 0) given by with p, q, r ∈ ℕ, p, q, r ≥ 2 and (p, q) = 1. Such a germ has an isolated singularity at 0 and gives rise to a Milnor fibration . Moreover, it is known that the link L_F is a Seifert manifold and that it is always homeomorphic to the link of a complex singularity. However, we prove that in almost all the cases the open-book decomposition of 𝕊⁵ given by the Milnor fibration of F cannot come from the Milnor fibration of a complex singularity in ℂ³.

We consider amalgamated unital full free products of the form A₁*_D A₂, where A₁, A₂ and D are finite-dimensional C*-algebras and there are faithful traces on A₁ and A₂ whose restrictions to D agree. We provide several conditions on the matrices of partial multiplicities of the inclusions D ↪ A₁ and D ↪ A₂ that guarantee that the C*-algebra A₁*_D A₂ is primitive. If the ranks of the matrices of partial multiplicities are one or all entries are 0 or ≥ 2, we prove that the algebra A₁*_D A₂ is primitive if and only if it has a trivial center.

We introduce augmented biracks and define a (co)homology theory associated to augmented biracks. The new homology theory extends the previously studied Yang–Baxter homology with a combinatorial formulation for the boundary map and specializes to N-reduced rack homology when the birack is a rack. We introduce augmented birack 2-cocycle invariants of classical and virtual knots and links and provide examples.

Given a regular curve in Minkowski spacetime, we describe necessary and sufficient conditions for this curve to support a family of pairwise-disjoint crooked planes. Using this criterion, we describe crooked foliations along orbit curves of one-parameter groups of Lorentzian isometries.

The geography of log models refers to the decomposition of the set of effective adjoint divisors into the polytopes defined by the resulting models obtained by the log minimal model program. We will describe the geography of log models in terms of the asymptotic base loci and Zariski decompositions of adjoint divisors. As an application, we prove some structure theorems on partially ample cones, thereby giving a partial answer to a question of B. Totaro.

Let denote the class of all analytic functions f in the open unit disk with the normalizations f(0) = f′(0) - 1 = 0 such that zf′(z)/f(z) lies in the interior of the left-half of the shifted lemniscate of Bernoulli . In this paper, we investigate the geometric properties of functions in this class and compute the -radius for functions belonging to several interesting classes.

The main objective in this paper is to obtain the existence results for bounded and unbounded solutions of some quasilinear elliptic systems. Related results as obtained here have been established recently in [C. O. Alves and A. R. F. de Holanda, Existence of blow-up solutions for a class of elliptic systems, Differ. Integral Eqs.26(1/2) (2013) 105–118]. Also, we present some references to give the connection between these types of problems with probability and stochastic processes, hoping that these are interesting for the audience of analysts likely to read this paper.