Algebras and differential equations with additive symmetry

Abstract

The set of all m-ary algebra structures on a given vector space affords, by the change of basis action, a representation of the general linear group. The invariants of a given subgroup are identified with those algebras admitting that subgroup as algebra automorphisms. Any finite dimensional representation of the additive group as automorphisms is obtained as the exponential of a nilpotent derivation. The latter can be embedded in the Lie algebra sl(2) so that the maximal vectors in an irreducible decomposition of the set of algebras as an sl(2) module are the invariants of the given action of the additive group. Dimension formulas and explicit bases are computed for the space of algebras with certain additive group actions. Employing the equivalence of the categories of m-ary algebras and systems of autonomous mth order homogeneous differential equations, the algebraic results are connected to the construction of first integrals and semi-invariants.