On connection between variational symmetries and algebraic structures

Abstract

In the work we present a rather general approach for finding connections between the symmetries of $B_u$-potentials, variational symmetries, and algebraic structures, Lie-admissible algebras and Lie algebras. In order to do this, in the space of the generators of the symmetries of the functionals we define such bilinear operations as $S$ $(\mathscr{S},\mathscr{T})$-product, $\mathscr{G}$-commutator, commutator. In the first part of the work, to provide a complete description, we recall needed facts on $B_u$-potential operators, invariant functionals and variational symmetries. In the second part we obtain conditions, under which $(\mathscr{S},\mathscr{T})$-product, $\mathscr{G}$-commutator, commutator of symmetry generators of $B_u$-potentials are also their symmetry generators. We prove that under some conditions $(\mathscr{S},\mathscr{T})$-product turns the linear space of the symmetry generators of $B_u$-potentials into a Lie-admissible algebra, while $\mathscr{G}$-commutator and commutator do into a Lie algebra. As a corollary, similar results were obtained for the symmetry generators of potentials, $B_u\equiv I$, where the latter is the identity operator. Apart of this, we find a connection between the symmetries of functionals with Lie algebras, when they have bipotential gradients. Theoretical results are demonstrated by examples.