Parity-independent 9-j q S symbols for the deformed superalgebra Uq(osp(1|2)) from Racah's and Biedenharn-Elliott's sum rules

摘要

A preliminary re-phasing of the sq6-j symbols is carried out, thus transforming them into “standard" parity-independent 6-j q S symbols. In terms of these 6-j q S, Uq(osp(1|2)) pseudo-orthogonality relation, Racah sum rule and Biedenharn-Elliott identity are re-written, evidencing a perfect similarity with the analogous formulas related to osp(1|2). From the combination of Racah and Biedenharn-Elliott sum rules, we define easily 9-j q S symbols, invariant under transposition. Their permutational symmetries, the list of 9-j q S with an argument zero as a function of a single 6-j q S symbol are presented, as well as the pseudo-orthogonality relation. Numerous summation formulas involving products of 9-j q S, or 9-j q S and 6-j q S symbols are derived. In the multiplicative factors involving powers of q--deformation parameter related to Kulish numbers--the dependence is found to be the same as for suq(2), with the formal replacement of q into q-1. The phases found are quite similar to those known for osp(1|2). Finally a simple mnemonic-like rule is presented for gathering the summation formulas with generic 6-j, 9-j symbols for Uq(osp(1|2)), osp(1|2), suq(2) and su(2). A possible unified approach is outlined.