Wolfes model aka G2/I6-rational integrable model: g(2), g(3) hidden algebras and quartic polynomial algebra of integrals

Abstract

One-dimensional 3-body Wolfes model with 2- and 3-body interactions also known as G2/I6-rational integrable model of the Hamiltonian reduction is exactly-solvable and superintegrable. Its Hamiltonian H and two integrals I1,I2, which can be written as algebraic differential operators in two variables (with polynomial coefficients) of the 2nd and 6th orders, respectively, are represented as non-linear combinations of g(2) or g(3) (hidden) algebra generators in a minimal manner. By using a specially designed MAPLE-18 code to deal with algebraic operators it is found that (H,I1,I2,I12≡[I1,I2]) are the four generating elements of the quartic polynomial algebra of integrals. This algebra is embedded into the universal enveloping algebra g(3). In turn, 3-body/A2-rational Calogero model is characterized by cubic polynomial algebra of integrals, it is mentioned briefly.