Topics in hidden symmetries. IV

Abstract

This note being devoted to some aspects of the inverse problem of representation theory explicates the links between researches on the Sklyanin algebras and the author's (based on the noncommutative geometry) approach to the setting free of hidden symmetries in terms of "the quantization of constants". Namely, the Racah-Wigner algebra for the Sklyanin algebra is constructed. It may be considered as a result of the quantization of constants in the Racah-Wigner algebra for the Lie algebra $sl(2,C)$. The Racah-Wigner algebra for the Sklyanin algebra is an example of the noncommutative weighted shift operator algebras (NWSO-algebras), which generalize the mho-algebras introduced by the author earlier. If the Sklyanin algebra is interpreted as an algebra of anomalous spins then the Racah-Wigner algebra for it may be regarded as an enlargement of the Sklyanin algebra by operators of the anomalous spin-spin interaction (of tensor type).