Article

Ufa Mathematical Journal
Volume 13, Number 1, pp. 46-55

On connection between variational symmetries and algebraic structures

Budochkina S.A.

DOI:10.13108/2021-13-1-46

In the work we present a rather general approach for finding connections between the symmetries of $B_u$-potentials, variational symmetries, and algebraic structures, Li-admissible algebras and Li algebras. In order to do this, in the space of the generators of the symmetries of the functionals we define such bilinear operations as $({{S}},{ T})$-product, ${ G}$-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 $({ S},{ T})$-product, ${ G}$-commutator, commutator of symmetry generator of $B_u$-potentials are also its symmetry generator. We prove that under some conditions $({ S},{ T})$-product turns the linear space of the symmetry generators of $B_u$-potential into a Li-admissible algebra, while ${ G}$-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 mapping. 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.