Modular transformations for tensor categories
type: Item
instance of: scholarly article
title: Modular transformations for tensor categories
zbMATH Open document ID: 0823.18003
DOI: 10.1016/0022-4049(94)00045-K
published in: Journal of Pure and Applied Algebra
publication date: 1995-06-08T00:00:00Z
review text: This fine paper follows on from the author's [Tangles and Hopf algebras in braided categories, ibid. 98, No. 3, 245-278 (1995; see the preceding review)]. It begins by showing the existence and uniqueness of an integral for each Hopf algebra in an abelian autonomous braided category \({\mathcal C}\). The Hopf algebra \(F\) of functions on \({\mathcal C}\) is defined to be the coend of the internal hom functor of \({\mathcal C}\). The universal enveloping algebra \({\mathbf u}\) is a specific Hopf algebra defined as a subalgebra of the left dual of \(F\); there is a canonical epimorphism \(j : F \to {\mathbf u}\). The integral of a tortile tensor category is a certain morphism from the unit object \(I\) to \({\mathbf u}\), while the Fourier transform is a specific endomorphism \(S\) of \({\mathbf u}\). The terminology is explained: certain modular relations satisfied by the Fourier transform on \(L^ 2 (\mathbb{R})\) are shown to hold for \(S\).
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 18E10
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 57T05
Mathematics Subject Classification ID: 18D35
zbMATH DE Number: 762342
zbMATH Keywords: monoidal category
zbMATH Keywords: rigid
zbMATH Keywords: integral
zbMATH Keywords: Hopf algebra
zbMATH Keywords: tortile tensor category
zbMATH Keywords: Fourier transform
author: Volodymyr Lyubashenko
reviewed by: Ross H. Street
MaRDI profile type: Publication
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full work available at URL: 0022-4049(94)00045-k
OpenAlex ID: W2002891627
description: scientific article; zbMATH DE number 762342