Topics in Hidden Symmetries. Iv.

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). 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). 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. This paper being a continuation of the previous three parts [1-3] as illustrates the general ideology presented in the review [4] as explicates its new features. The subject is new links between a research activity on the Sklyanin algebras [5,6] (see also [7]) and the author’s (based on the noncommutative geometry) approach to the setting free of hidden symmetries [3:§1] (see also [8,1]) in terms of ”the quantization of constants” [4:§3;3]. More concretely, we construct the Racah–Wigner algebra for the Sklyanin algebra, which may be considered also as a result of the quantization of constants in the Racah–Wigner algebra for the Lie algebra so(3,C) ≃ sl(2,C). Formally, the Racah–Wigner algebra for the Sklyanin algebra is one of examples of the noncommutative weighted shift operator algebras (NWSO–algebras), which generalize the mho–algebras introduced by the author earlier [1:§2] (see also [3:§1.4]). Because the Sklyanin algebra is a very interesting object, deeply related to many algebraic structures (see e.g.[9,10]), and the Racah–Wigner algebra for the Lie algebra sl(2,C) [8;11:§2.2] (see also [3:§§1,2;2] and cf.[12:§1.2]) is not less intriguing, their ”hybrid” will be undoubtly of a certain value.