6J Symbols Duality Relations

TL;DR

This paper explores the algebraic structures, specifically double crossproduct and bicrossproduct Hopf algebras, that underlie the Fourier transformation identities of 6j symbols in quantum groups and related algebraic systems.

Contribution

It identifies the Hopf algebra structures responsible for the duality relations of 6j symbols, extending understanding to various algebraic and quantum group contexts.

Findings

Hopf algebra structures explain 6j symbol duality relations

Analysis of examples including group algebras and quantum doubles

Connections established between algebraic structures and Fourier identities

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.