Note on 2d binary operadic harmonic oscillator

It is explained how the time evolution of the operadic variables may be introduced. As an example, a 2-dimensional binary operadic Lax representation of the harmonic oscillator is found.


Introduction
It is well known that quantum mechanical observables are linear operators, i.e the linear maps V → V of a vector space V and their time evolution is given by the Heisenberg equation. As a variation of this one can pose the following question [7]: how to describe the time evolution of the linear algebraic operations (multiplications) V ⊗n → V . The algebraic operations (multiplications) can be seen as an example of the operadic variables [2,3,4,5].
When an operadic system depends on time one can speak about operadic dynamics [7]. The latter may be introduced by simple and natural analogy with the Hamiltonian dynamics. In particular, the time evolution of operadic variables may be given by operadic Lax equation. In [8] it was shown how the dynamics may be introduced in 2d Lie algebra. In the present paper, an operadic Lax representation for harmonic oscillator is constructed in general 2d binary algebras.

Operad
Let K be a unital associative commutative ring, and let C n (n ∈ N) be unital K-modules. For f ∈ C n , we refer to n as the degree of f and often write (when it does not cause confusion) f instead of deg f . For example, (−1) f . = (−1) n , C f . = C n and • f . = • n . Also, it is convenient to use the reduced degree |f | . = n − 1. Throughout this paper, we assume that ⊗ . = ⊗ K .
Definition 2.1 (operad (e.g [2,3])). A linear (non-symmetric) operad with coefficients in K is a sequence C . = {C n } n∈N of unital K-modules (an N-graded K-module), such that the following conditions are held to be true.
(1) For 0 ≤ i ≤ m − 1 there exist partial compositions (3) Unit I ∈ C 1 exists such that In the second item, the first and third parts of the defining relations turn out to be equivalent.
Example 2.2 (endomorphism operad [2]). Let V be a unital K-module and E n Therefore, algebraic operations can be seen as elements of an endomorphism operad.
Just as elements of a vector space are called vectors, it is natural to call elements of an abstract operad operations. The endomorphism operads can be seen as the most suitable objects for modelling operadic systems.
Definition 3.2 (Gerstenhaber brackets [2,3]). The Gerstenhaber brackets [·, ·] are defined in Com C as a graded commutator by The commutator algebra of Com C is denoted as One can prove that Com − C is a graded Lie algebra. The Jacobi identity reads Assume that K . = R and operations are differentiable. The dynamics in operadic systems (operadic dynamics) may be introduced by the

Operadic harmonic oscillator
Consider the Lax pair for the harmonic oscillator: it is easy to check that the Lax equatioṅ If µ is a homogeneous operadic variable one can use the above Hamilton's equations to obtain dµ dt = ∂µ ∂q dq dt + ∂µ ∂p dp dt = p ∂µ ∂q − ω 2 q ∂µ ∂p Therefore, the linear partial differential equation for the operadic variable µ(q, p) reads By integrating one gains sequences of operations called the operadic (Lax representations of ) harmonic oscillator.

Example
Let A . = {V, µ} be a binary algebra with operation xy . = µ(x ⊗ y). We require that µ = µ(q, p) so that (µ, M ) is an operadic Lax pair, i.e the operadic Lax equatioṅ Let x, y ∈ V . By assuming that |M | = 0 and |µ| = 1, one has Therefore, one has For the harmonic oscillator, define its auxiliary functions A ± and D ± by Then one has the following M ) is a 2-dimensional binary operadic Lax pair of the harmonic oscillator.
Then it follows from Lemma 5.1 that the 2-dimensional binary operadic Lax equations read C β Γ β α = 0, α = 1, . . . , 8 Since the parameters C β are arbitrary, the latter constraints imply Γ = 0. Thus one has to consider the following differential equations