Modular transformations for tensor categories (Q1892308)

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 \(S\).