Generalized and Quasi-Localizations of Braid Group Representations

Abstract

Abstract. We develop a theory of localization for braid group representations associated with objects in braided fusion categories and, more generally, to YangBaxter operators in monoidal categories. The essential problem is to determine when a family of braid representations can be uniformly modelled upon a tensor power of a fixed vector space in such a way that the braid group generators act "locally". Although related to the notion of (quasi-)fiber functors for fusion categories, remarkably, such localizations can exist for representations associated with objects of non-integral dimension. We conjecture that such localizations exist precisely when the object in question has dimension the square-root of an integer and prove several key special cases of the conjecture.

Dimension as a quantum statistic and the classification of metaplectic categories

“…It is believed that this relationship holds more generally: 45,25]). Any simple X ∈ C is (generalized or quasi-)localizable if, and only if dim(X) 2 ∈ Z.…”

“…We say that X ∈ C is localizable if there is a braided vector space (R, V ) and injective algebra maps τ n : Cρ X (B n ) → End(V ⊗n ) such that, for all n, ρ R = τ n • ρ X . That is, the following diagram commutes: Two slightly less restrictive notions of localizability are studied in [25], namely (k, m)generalized localizations and quasi-localizations. Under some assumptions, localizability is known to be determined by dimension: 45,25]).…”

“…That is, the following diagram commutes: Two slightly less restrictive notions of localizability are studied in [25], namely (k, m)generalized localizations and quasi-localizations. Under some assumptions, localizability is known to be determined by dimension: 45,25]). Let X ∈ C be a simple object.…”

Invariants of Links From the Generalized Yang–baxter Equation

“…A (2, 3, 2) gYB-operator is given in [9] and a (2, 3, 1) gYB-operator is in [3]. Three families of variations of the latter are obtained in [2] and we will call those as (2, 3, 1) gYB-operator of type I, II, and III in the following Sec.…”

“…If there is a (generalized) localization in the sense of [3,8], then some enhancement should exist and it is reasonable to expect that the corresponding invariant recovers the one defined directly from the category. In [3], it is shown that SU(3) 3 theory has no 4 × 4 ordinary localization which corresponds to the usual YB-operator, but does have an 8 × 8 generalized localization which corresponds to gYB-operator. No 4 × 4 unitary YB-operator can yield the invariant discussed in [7] (and identified in [6]), and it seems unlikely that any ordinary YB-operator can.…”