Abstract. For a pivotal finite tensor category C over an algebraically closed field k, we define the algebra CF(C) of class functions and the internal character ch(X) ∈ CF(C) for an object X ∈ C by using an adjunction between C and its monoidal center Z(C). We also develop the theory of integrals and the Fourier transform in a unimodular finite tensor category by using the same adjunction. Our main result is that the map ch : Gr k (C) → CF(C) given by taking the internal character is a well-defined injective homomorphism of kalgebras, where Gr k (C) is the scalar extension of the Grothendieck ring of C to k. Moreover, under the assumption that C is unimodular, the map ch is an isomorphism if and only if C is semisimple.As an application, we show that the algebra Gr k (C) is semisimple if C is a non-degenerate pivotal fusion category. If, moreover, Gr k (C) is commutative, then we define the character table of C based on the integral theory. It turns out that the character table is obtained from the S-matrix if C is a modular tensor category. Generalizing corresponding results in the finite group theory, we prove the orthogonality relations and the integrality of the character table.
For a braided finite tensor category C with unit object 1 ∈ C, Lyubashenko considered a certain Hopf algebra F ∈ C endowed with a Hopf pairing ω : F ⊗ F → 1 to define the notion of a 'non-semisimple' modular tensor category. We say that C is non-degenerate if the Hopf pairing ω is nondegenerate. In this paper, we show that C is non-degenerate if and only if it is factorizable in the sense of Etingof, Nikshych and Ostrik, if and only if its Müger center is trivial, if and only if the linear map Hom C (1, F) → Hom C (F, 1) induced by the pairing ω is invertible. As an application, we prove that the category of Yetter-Drinfeld modules over a Hopf algebra in C is non-degenerate if and only if C is.
Sialic acid-binding immunoglobulin-like lectin-8 (Siglec-8) promotes the apoptosis of eosinophils and inhibits FceRI-dependent mediator release from mast cells. We investigated the genetic association between sequence variants in Siglec-8 and diagnosis of asthma, total levels of serum IgE (tIgE), and diagnosis of eosinophilic esophagitis (EE) in diverse populations. The effect of sequence variants on Siglec-8 glycan ligand-binding activity was also examined. Significant association with asthma was observed for SNP rs36498 (odds ratios (OR), 0.69, P¼8.8Â10 À5 ) among African Americans and for SNP rs10409962 (Ser/Pro) in the Japanese population (OR, 0.69, P¼0.019). Supporting this finding, we observed association between SNP rs36498 and current asthma among Brazilian families (P¼0.013). Significant association with tIgE was observed for SNP rs6509541 among African Americans (P¼0.016), and replicated among the Brazilian families (P¼0.02). In contrast, no association was observed with EE in Caucasians. By using a synthetic polymer decorated with 6¢-sulfo-sLe x , a known Siglec-8 glycan ligand, we did not find any differences between the ligand-binding activity of HEK293 cells stably transfected with the rs10409962 risk allele or the WT allele. However, our association results suggest that the Siglec8 gene may be a susceptibility locus for asthma.
Electrophoretickaryotypes of Japanese wine yeasts whose strain identities had been confused were examined by using pulsed field gel electrophoresis (PFG). PFG pattern of KW-1 was different from that of OC-2, which had been described to be the original strain of KW-1. KW-2, which was reported to be constructed by mating between haploids from K-6 and KW-1, showed several chromosomal bands which were not observed in K-6, KW-1 and OC-2.
Abstract. In this paper, we introduce the notion of the pivotal cover C piv of a left rigid monoidal category C to develop a theoretical foundation for the theory of Frobenius-Schur (FS) indicators in "non-pivotal" settings. For an object V ∈ C piv , the (n, r)-th FS indicator νn,r(V) is defined by generalizing that of an object of a pivotal monoidal category. This notion gives a categorical viewpoint to some recent results on generalizations of FS indicators.Based on our framework, we also study the FS indicators of the "adjoint object" in a finite tensor category, which can be considered as a generalization of the adjoint representation of a Hopf algebra. The indicators of this object closely relate to the space of endomorphisms of the iterated tensor product functor.
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