Operadic representations of harmonic oscillator in some 3d algebras

Abstract

It is explained how the time evolution of the operadic variables may be introduced by using the operadic Lax equation. The operadic Lax representations for the harmonic oscillator are constructed in 3-dimensional binary anti-commutative algebras. As an example, an operadic Lax representation for the harmonic oscillator in the Lie algebra $\mathfrak{sl}(2)$ is constructed.