Racah Sum Rule and Biedenharn-Elliott Identity for the Super-Rotation $6-j$ symbols

TL;DR

This paper extends the Racah sum rule and Biedenharn-Elliott identity, originally for SO(3) rotation algebra, to the super-rotation osp(1|2) superalgebra, highlighting structural similarities and sign complexities.

Contribution

It demonstrates the extension of key recoupling identities from classical to super-rotation algebra, revealing structural parallels and sign intricacies.

Findings

Sum rules are structurally similar for both algebras

Sign differences are more complex in super-rotation case

Extension broadens understanding of superalgebra recoupling

Abstract

It is shown that the well known Racah sum rule and Biedenharn-Elliott identity satisfied by the recoupling coefficients or by the $6-j$ symbols of the usual rotation $SO(3)$ algebra can be extended to the corresponding features of the super-rotation $osp(1|2)$ superalgebra. The structure of the sum rules is completely similar in both cases, the only difference concerns the signs which are more involved in the super-rotation case.