On observable algebras of a class of associative mechanical systems (Q1070329)

It is assumed that the observables of a physical system are the elements of the universal covering \(U(\mathfrak g)\) (or its factor algebra) of some Lie algebra \(\mathfrak g\), and the universal covering \(U(\mathfrak g)\) is assumed to be invariant with respect to the adjoint representation of the Lie group \(G\), for which \(\mathfrak g\) is the Lie algebra. In the first part of the paper a classification is made of the associative algebras of functions that are invariant with respect to the adjoint representation of \(G\), where \(G\) is the group of affine transformations of the real line. The second part of the paper is devoted to the application of the obtained classification to the problem of describing the possible associative Hamiltonian mechanical systems.