6J Symbols Duality Relations

Abstract

It is known that the Fourier transformation of the square of (6j) symbols has a simple expression in the case of su(2) and U_q(su(2)) when q is a root of unit. The aim of the present work is to unravel the algebraic structure behind these identities. We show that the double crossproduct construction H_1\bowtie H_2 of two Hopf algebras and the bicrossproduct construction H_2^{*}\lrbicross H_1 are the Hopf algebras structures behind these identities by analysing different examples. We study the case where D= H_1\bowtie H_2 is equal to the group algebra of ISU(2), SL(2,C) and where D is a quantum double of a finite group, of SU(2) and of U_q(su(2)) when q is real.

28 pages, 2 figures

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• Refined Chern-Simons Theory in Genus Two
• Quantum geometry from higher gauge theory
• BTZ Black Hole Entropy and the Turaev-Viro model
• Classical r-matrices via semidualisation
• A Gauge-Independent Mechanism for Confinement and Mass Gap: Part I -- The General Framework