On Vafa’s theorem for tensor categories

By Pavel Etingof. 2002

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In this note we prove two main results. 1. In a rigid braided finite tensor category over C (not necessarily semisimple), some power of the Casimir element and some even power of the braiding is unipotent. 2. In a (semisimple) modular category, the twists are roots of unity dividing the algebraic integer D^{5/2}, where D is the global dimension of the category (the sum of squares of dimensions of simple objects). Both results generalize Vafa's theorem, saying that in a modular category twists are roots of unity, and square of the braiding has finite order. We also discuss the notion of the quasi-exponent of a finite rigid tensor category, which is motivated by results 1 and 2 and the paper math/0109196 of S.Gelaki and the author.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b9eac2573993798f9d1e209babc612c047735680

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Citations in the corpus

Frobenius–Schur indicators and exponents of spherical categories (2005, result): “s dim(H)4. The techniques in this case are close to those of Etingof and Gelaki in [EG02]. Special care has to be taken to deal with the nontrivial associativity isomorphisms (which can be avoided in [Eti] by using categorically deﬁned determinants, at the cost of a higher bound). Our bound is highe”
TFT construction of RCFT correlators: III: simple currents (2004, background): “3 The numbers i are indeed phases, and even roots of unity, see e.g. [ 37 ].”
Torus bundles not distinguished by TQFT invariants (2011, methodology): “Finally, the so-called Vafa Theorem (see Vafa [75] and Etingof [20]) shows that the anomaly 3 of the TQFT associated to D.”
Tensor categories: a selective guided tour (2008, background): “) – The elements of T matrix are roots of unity, and the elements of S are cyclotomic integers [36, 78].”
Finite dimensional quasi-Hopf algebras with radical of codimension 2 (2003, unknown): “Pdim(X), called the Frobenius-Perron (FP) dimension of X, and the Frobenius-Perron (FP) dimension of C is deﬁned by FPdim(C) = X X∈Irr(C) FPdim(X)FPdim(PX), where PX denotes the projective cover of X [E]. By Proposition 2.6 in [EO], C is equivalent to Rep(H), H a ﬁnite dimensional quasi-Hopf algebra”