Automorphic algebras of dynamical systems and generalised In¨on¨u-Wigner contractions

Abstract

Lie algebras a with a complex underlying vector space V are studied that are automorphic with respect to a given linear dynamical system on V , i.e., a 1-parameter subgroup Gt ⊂ Aut(a) ⊂ GL(V ). Each automorphic algebra imparts a Lie algebraic structure to the vector space of trajectories of the group Gt. The basics of the general structure of automorphic algebras a are described in terms of the eigenspace decomposition of the operatorM ∈ der(a) that determines the dynamics. Symmetries encoded by the presence of nonabelian automorphic algebras are pointed out connected to conservation laws, spectral relations and root systems. It is shown that, for a given dynamics Gt, automorphic algebras can be found via a limit transition in the space of Lie algebras on V along the trajectories of the group Gt itself. This procedure generalises the well-known Inönü-Wigner contraction and links adjoint representations of automorphic algebras to isospectral Lax representations on gl(V ). These results can be applied to physically important symmetry groups and their representations, including classical and relativistic mechanics, open quantum dynamics and nonlinear evolution equations. Simple examples are given.