Categorical Morita Equivalence for Group-Theoretical Categories

By D. Venku Naidu. 2006

73 citations recorded by the source corpus.

A finite tensor category is called pointed if all its simple objects are invertible. We find necessary and sufficient conditions for two pointed semisimple categories to be dual to each other with respect to a module category. Whenever the dual of a pointed semisimple category with respect to a module category is pointed, we give explicit formulas for the Grothendieck ring and for the associator of the dual. This leads to the definition of categorical Morita equivalence on the set of all finite groups and on the set of all pairs (G, ω), where G is a finite group and ω ∈ H 3(G, k ×). A group-theoretical and cohomological interpretation of this relation is given. A series of concrete examples of pairs of groups that are categorically Morita equivalent but have nonisomorphic Grothendieck rings are given. In particular, the representation categories of the Drinfeld doubles of the groups in each example are equivalent as braided tensor categories and hence these groups define the same modular data.

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

View Paper: https://www.semanticscholar.org/paper/2c7b8893c880656c58cc2273aab05d705d357aa4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The Picard crossed module of a braided tensor category (2012, result): “The following lemma is a special case of the result proved in [Naidu 2007]:”
Universal Non‐Invertible Symmetries (2022, background): “If two 1-categories C and D are Morita equivalent, then Mod(C) and Mod(D) are the same (more precisely there is a 2-equivalence between them)[27, 41].”
Categorical Lagrangian Grassmannians and Brauer-Picard groups of pointed fusion categories (2013, background): “in H2(L′, k×). Fix a subgroup L of G. Let E denote the group of isomorphism classes of right VecG-module autoequivalences of M(L, µ) isomorphic to the identity as an additive functor. It follows from [N] that there is a group isomorphism (8) ι : E → Lb : F → ιF 6 DMITRI NIKSHYCH AND BRIANNA RIEPEL s”
GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES (2007, background): “C ∼= Z(Vecω G) as a braided fusion category. Indeed, it follows from [EG1] that there exist non-isomorphic ﬁnite groups G 1,G 2 such that Z(VecG1) ∼= Z(VecG 2) as braided fusion categories. (See also [N].) Theorem 4.8. Let C be a modular category such that dim(C) = n2, n ∈ Z+, and such that ξ(C) = 1”
On gauging finite subgroups (2017, background): “When D = 2 this result was known in the mathematical literature [29,30] for some time.”
On finite symmetries and their gauging in two dimensions (2017, background): “There is a theorem by Naidu, Nikshych [29, 30] and Uribe [31] determining exactly when this happens, and if so, explicitly the form of G′ and α′.”
Theta, time reversal and temperature (2017, background): “19 An analogous phenomenon concerning the bonus symmetries in 2d orbifolds is described in the mathematical literature in [54].”
Lagrangian Subcategories and Braided Tensor Equivalences of Twisted Quantum Doubles of Finite Groups (2007, methodology): “ that G 6∼= G′ in general. There is a canonical subcategory L ({1},1) ∼= Rep(G) corresponding to the forgetful functor Rep(Dω(G)) ∼= Z(Vecω G ) → Vec ω G . We have L ({1},1) ∩ L (H,B) ∼= Rep(G/H). In [Na] the ﬁrst named author classiﬁed indecomposable Vecω G -module categories M with the property th”
The Quantum Double Model with Boundary: Condensations and Symmetries (2010, methodology): “This question was first answered by Naidu and Nikshych [11,12] based on the classification of Lagrangian subcategories of Z(G).”
Non-group-theoretical semisimple Hopf algebras from group actions on fusion categories (2007, background): “where N is a normal Abelian subgroup of G such that !|N ×N ×N is trivial and � ∈ H 2 (N, k × ) is a G-invariant cohomolo gy class, see [Na].”