The monoidal center construction and bimodules

By Peter Schauenburg. 2001

82 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/710ac8e949871d7a58e0f07553fa28b598b78f2e

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Citations in the corpus

Anyon condensation and tensor categories (2013, methodology): “tensor category C, then we have C⊠Cloc A ≃Z(CA). Moreover, since Bis aconnected separable algebra inCA, it isknown that CB|B, which can be naturally identiﬁed with (CA)B|B, is Morita equivalent to CA [S] and we have Z(CB|B) ≃Z(CA) as modular tensor categories [M1][ENO2]. Therefore, the results given”
The Witt group of non-degenerate braided fusion categories (2010, background): “ functors of right multiplication by objects of Z(A). Combined, these identiﬁcations yield a canonical equivalence of braided categories (12) Z(A) −→ Z∼ (A∗ M). This result is due to Schauenburg, see [Sch]. 3. Etale algebras and central functors´ 3.1. Etale algebras in braided fusion categories.´ De”
Lagrangian Subcategories and Braided Tensor Equivalences of Twisted Quantum Doubles of Finite Groups (2007, background): “. That the “if” part is true for all fusion categories was ﬁrst observed by M. Mu¨ger in [Mu1, Remark 3.18]. This follows from the deﬁnition of weak Morita equivalence and a theorem of P. Schauenburg [S]. See also [N, O1, EO]. For the “only if” part, let (G1, ω1),(G2, ω2) be two pairs of groups and ”
Bicategories for Boundary Conditions and for Surface Defects in 3-d TFT (2012, background): “ the left, or, alternatively, with the functors of tensoring with an object of Z(EndA(M)) from the right. Comparing the two descriptions of these functors gives the following result: Proposition 2.7. [Sc] For any module M over a fusion category A there is a canonical equivalence Z(A) −→ Z≃ (End A(M)”
Drinfeld Center and Representation Theory for Monoidal Categories (2015, background): “emma 5.11. The pair (mr,ml) = (ι⊗mQ,(ι⊗mQ)(cQ ⊗ι)) deﬁnes the structure of a Q-bimodule on X⊗Q. Proof. Let us concentrate on the ∗-compatibility of ml, since this is the only new property compared to [Sch01]. We have ml∗ = (c∗ Q⊗ιQ)(ιX ⊗w) and we want to show that this is equal to (ιQ⊗ml)(wv⊗ιX ⊗ιQ)”
Tensor categories: a selective guided tour (2008, background): “(This is an immediate corollary of the definition of ≈, combined with [240].”
Frobenius–Schur indicators and exponents of spherical categories (2005, methodology): “t using semisimplicity: By [Sch02] the category of modules over the double D(H) is isomorphic to the monoidal category HCH of H-H-bicomodules in the monoidal category C = H-modfin-H of Hbimodules. By [Sch01] the center of this bicomodule category is equivalent to the center of the underlying categor”
On the Gauge Equivalence of Twisted Quantum Doubles of Elementary Abelian and Extra-Special 2-Groups (2006, background): “ras among twisted quantum doubles Dω(G) where Gis a group deﬁned as an extension of Gby an abelian group and the 3-cocycle ω∈Z3(G,C∗) is inﬂated from a 3-cocycle on G. Several results of Schauenburg ([S1], [S2]) play a signiﬁcant rˆole in the proof. 2 As long as l ≥2, the group of elements of H3(Q,C”
Fusion categories and homotopy theory (2009, unknown): “This result is due to Schauenburg; see [35].”