Quadratic conservatives of linear symplectic system (Q1112390)

This paper addresses the problem of determining algebraically which quadratic forms \(f_ S(x)=x^ TSx\) (with \(S\) symmetric) are conserved along any solution of the linear recurrence on \({\mathbb R}^ N\), \(x_{t+1}=Ax_ t\), where \(A\) is an element of the symplectic group \(\text{Sp}(N,{\mathbb R})\). Quadratic conservatives (i.e., conserved quadratic forms) form a Lie algebra with respect to the Poisson bracket. The author shows that, in the linear space of all matrices commuting with \(A\), one of the subspaces is isomorphic (as a Lie algebra) to the space of quadratic conservatives of the linear recurrence. Motivating the author's work is the desire to find conservation laws for a discrete economic growth model.