Traces on module categories over fusion categories

Schaumann, Gregor

doi:10.1016/j.jalgebra.2013.01.013

Cited by 41 publications

(51 citation statements)

References 23 publications

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“…So let us first assume that the squared antipode is the conjugation by a trivial group-like element, i.e., S 2 (x) = gxg −1 , where g = yS(y) −1 and y ∈ A s ∼ = A is an invertible element of the source base of H (this holds in all the examples we know). In this case, following Schaumann ([Sch1]), we will say that the pivotal structure on C defined by g is matched to the module category M; we show that our definition is equivalent to that of [Sch1]. In the pseudounitary case it is shown in [N2] that this is automatic.…”

Section: Introductionmentioning

confidence: 89%

“…Acknowledgements. P. Etingof thanks D. Nikshych and V. Ostrik for useful discussions and comments on the draft of this paper, and in particular to V. Ostrik for providing the reference [Sch1]. He is also very grateful to G. Schaumann for comments on the draft of the paper and for contributing an appendix.…”

Section: Introductionmentioning

confidence: 93%

“…We start with giving a straightforward generalization of the results of [Sch1] to the setting of non-semisimple tensor categories in any characteristic. So this subsection is essentially an exposition of the results of [Sch1] in a slightly more general setting using a somewhat different approach.…”

Section: Pivotal Structures Matched To a Module Categorymentioning

confidence: 99%

“…In the pseudounitary case it is shown in [N2] that this is automatic. In general, in the matched case it follows from [Sch1] that the characteristic polynomial of S 2 is still given by formula (1), but with m i now being nonzero numbers such that for all r, i we have j∈I N j ri m j = dim(X r )m i , where dim(X i ) is the dimension of X i in the pivotal structure defined by g (i.e., N r m = dim(X r )m, where N r := (N j ri ) and m = (m i )). More precisely, the numbers m i are uniquely determined up to scaling (as the dimension character X → dim(X) occurs with multiplicity one in the module Gr(M) ⊗ k over Gr(C) ⊗ k), and are all nonzero.…”

Section: Introductionmentioning

confidence: 99%

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Eigenvalues of the squared antipode in finite dimensional weak Hopf algebras

Etingof¹,

Schaumann²

2019

Tensor Categories and Hopf Algebras

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We extend Schaumann's theory of pivotal structures on fusion categories matched to a module category and of module traces to the case of non-semisimple tensor categories, and use it to study eigenvalues of the squared antipode S 2 in weak Hopf algebras. In particular, we diagonalize S 2 for semisimple weak Hopf algebras in characteristic zero, generalizing the result of Nikshych in the pseudounitary case. We show that the answer depends only on the Grothendieck group data of the pivotalizations of the categories involved and the global dimension of the fusion category (thus, all eigenvalues belong to the corresponding number field). On the other hand, we study the eigenvalues of S 2 on the nonsemisimple weak Hopf algebras attached to dynamical quantum groups at roots of 1 defined by D. Nikshych and the author in 2000, and show that they depend nontrivially on the continuous parameters of the corresponding module category. We then compute these eigenvalues as rational functions of these parameters. The paper also contains an appendix by G. Schaumann discussing the connection between our generalization of module traces and the notion of an inner product module category introduced in [Sch2].

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Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 93%

Section: Pivotal Structures Matched To a Module Categorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Eigenvalues of the squared antipode in finite dimensional weak Hopf algebras

Etingof¹,

Schaumann²

2019

Tensor Categories and Hopf Algebras

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“…These analogues of module categories involve both structure on M and compatibility of that structure with the module action. In the semisimple pivotal setting Schaumann [Sch13] showed that the appropriate structure is a choice of trace on M. In §3.4, we recall the de nitions of planar algebra, unitary dual functors, and unitary pivotal structure, and we explain the relationship between planar algebras and unitary pivotal fusion categories. In §3.5, we recall Schaumann's notion of trace and modify this notion to the unitary setting, we then de ne (unitary) pivotal modules, prove a (unitary) pivotal analogue of endofunctor embedding, and translate that into the desired graph planar algebra embedding theorem.…”

mentioning

confidence: 99%

The Asaeda–Haagerup fusion categories

Grossman

Izumi

Snyder

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. The classification of subfactors of small index revealed several new subfactors. The first subfactor above index 4, the Haagerup subfactor, is increasingly well understood and appears to lie in a (discrete) infinite family of subfactors where the Z/3Z symmetry is replaced by other finite abelian groups. The goal of this paper is to give a similarly good description of the Asaeda-Haagerup subfactor which emerged from our study of its Brauer-Picard groupoid. More specifically, we construct a new subfactor S which is a Z/4Z × Z/2Z analogue of the Haagerup subfactor and we show that the even parts of the Asaeda-Haagerup subfactor are higher Morita equivalent to an orbifold quotient of S. This gives a new construction of the Asaeda-Haagerup subfactor which is much more symmetric and easier to work with than the original construction. As a consequence, we can settle many open questions about the Asaeda-Haagerup subfactor: calculating its Drinfel'd center, classifying all extensions of the Asaeda-Haagerup fusion categories, finding the full higher Morita equivalence class of the AsaedaHaagerup fusion categories, and finding intermediate subfactor lattices for subfactors coming from the Asaeda-Haagerup categories. The details of the applications will be given in subsequent papers.

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On finite symmetries and their gauging in two dimensions

Bhardwaj¹,

Tachikawa²

2018

J. High Energ. Phys.

237

301

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It is well-known that if we gauge a Z n symmetry in two dimensions, a dual Z n symmetry appears, such that re-gauging this dual Z n symmetry leads back to the original theory. We describe how this can be generalized to non-Abelian groups, by enlarging the concept of symmetries from those defined by groups to those defined by unitary fusion categories. We will see that this generalization is also useful when studying what happens when a non-anomalous subgroup of an anomalous finite group is gauged: for example, the gauged theory can have non-Abelian group symmetry even when the original symmetry is an Abelian group. We then discuss the axiomatization of two-dimensional topological quantum field theories whose symmetry is given by a category. We see explicitly that the gauged version is a topological quantum field theory with a new symmetry given by a dual category. 2 Recently in [5], Gaiotto, Kapustin, Komargodski and Seiberg performed an impressive study of the phase structure of thermal 4d su(2) Yang-Mills theory. One important step in the analysis is the symmetry structure of the thermal system, which is essentially three-dimensional. As a dimensional reduction from 4d, the system has a Z 2 × Z 2 0-form symmetry and a Z 2 1-form symmetry, with a mixed anomaly. Then the authors gauged the Z 2 1-form symmetry, and found that the total 0-form symmetry is now D 8 . This D 8 was then used very effectively to study the phase diagram, but that part of their paper does not directly concern us here. Their analysis of turning an anomalous Abelian symmetry by gauging a non-anomalous subgroup into a non-Abelian symmetry is a 3d analogue of what we explain in 2d. See their Sec. 4.2, Appendix B and Appendix C. Clearly an important direction to pursue is to generalize their and our constructions to arbitrary combinations of possibly-higher-form symmetries in arbitrary spacetime dimensions, but that is outside of the scope of this paper.

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Traces on module categories over fusion categories

Cited by 41 publications

References 23 publications

Eigenvalues of the squared antipode in finite dimensional weak Hopf algebras

Eigenvalues of the squared antipode in finite dimensional weak Hopf algebras

The Asaeda–Haagerup fusion categories

On finite symmetries and their gauging in two dimensions

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