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

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.