Braided groups and quantum Fourier transform
The present paper is a good example of a close interaction between different categorical constructions related with quantum groups. In a previous paper the first author built the Hopf algebra \(F\) in any \(k\)- linear abelian ribbon tensor category \(\mathcal C\), and motivated from physics and low-dimensional topology defined invertible operators \({\mathcal S}_ -,{\mathcal T}\) on \(F\) obeying the modular identities \(({\mathcal S}_ - \circ{\mathcal T})^ 3 = \lambda {\mathcal S}_ -^ 2\), \({\mathcal S}_ -^ 2 = s^{-1}\), where \(s\) is the antipode of \(F\) and \(\lambda\) is a constant. The operator \(\mathcal S_ -\) plays the role of ``quantum Fourier transform''. As shown in the present paper \(F\) is indeed a realization of the general automorphism braided group \(\Aut({\mathcal C})\), obtained by the second author along the lines of a generalized Tannaka-Krein reconstruction theorem. For a special case when \(\mathcal C\) is the representation category of a quantum group \(H\) the algebra \(\underline H:=\Aut({\mathcal C})\) is a transmutation of \(H\). This allows the authors to define explicitly operators \({\mathcal S}_ -\) and \(\mathcal T\) on every finite-dimensional factorizable ribbon Hopf algebra. The class includes the finite-dimensional quantum groups \(u_ q(g)\) associated to complex simple Lie algebras. The example of \(u_ q(\text{sl}(2))\) at a root of unity is considered in detail, as well as an example relating to anyons. In the recent paper [\textit{A. Kempf} and \textit{S. Majid}, Algebraic \(q\)-integration and Fourier theory on quantum and braided spaces, J. Math. Phys. 35, No. 12, 6802-6837 (1994; Zbl 0826.17018)] convenient diagrammatic Fourier theory is developed for a general braided Hopf algebra. This includes the self-dual braided groups \(\underline H\) considered in the present paper. Connections of the Fourier transform with the low-dimensional topology are studied in future papers of the first author.
generalized Tannaka-Krein reconstruction
scientific article; zbMATH DE number 614243