Автоморфные алгебры динамических систем и обобщенные контракции Иненю-Вигнера

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.