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**Eugen PAAL ^{*} and J¨uri VIRKEPU**

Department of Mathematics, Tallinn University of Technology, Ehitajate tee 5, 19086 Tallinn, Estonia

- *Corresponding Author:
- Eugen PAAL

Department of Mathematics,

Tallinn University of Technology,

Ehitajate tee 5, 19086 Tallinn, Estonia

**E-mails:**[email protected], [email protected]

**Received date: ** December 12, 2007 **Accepted Date: **April 09, 2008

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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

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) 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 a 2-dimensional Lie algebra. In the present paper, an operadic Lax representation for the harmonic oscillator is constructed in general 2-dimensional binary algebras.

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 ο_{f} = ο_{n}. Also, it is convenient to use the *reduced* degree |f| = n − 1. Throughout this paper, we assume that

**Definition 2.1 **(operad (e.g [2,3])). A linear (non-symmetric) operad with coefficients in K is a sequence C= {C^{n}}_{n2N} of unital K-modules (an N-graded K-module), such that the following conditions are satisfied.

(1) For 0≤ i 0≤ m − 1 there exist partial compositions

*ο _{i} ∈ Hom(C^{m} C^{n},C^{m+n−1}), | ο_{i} | = 0*

(2) For all the composition (associativity) relations hold,

(3) Unit I ∈ C^{1} exists such that

I ο_{0}f = f = f ο_{i} I, 0≤ i ≤ |f|

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 Hom Define the partial compositions for

Then is an operad (with the unit ) called the endomorphism operad of V .

Thus, the 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.

**Definition 3.1 **(total composition [2,3]). The total composition •: is defined by

The pair ComC= {C, •} is called the composition algebra of C.

**Definition 3.2** (Gerstenhaber brackets [2,3]). The Gerstenhaber brackets [0≤, 0≤] are defined in ComC as a graded commutator by

The commutator algebra of ComC is denoted as Com^{−}C= {C, [0≤, 0≤]}. 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

**Definition 3.3** (operadic Lax pair [7]). Allow a classical dynamical system to be described by the evolution equations

An *operadic Lax pair* is a pair (L,M) of homogeneous operations L,M ∈ C, such that the above system of evolution equations is equivalent to the *operadic Lax equation*

Evidently, the degree constraints |M| = |L| = 0 give rise to ordinary Lax pair [6,1].

Consider the Lax pair for the harmonic oscillator:

Since the Hamiltonian is

it is easy to check that the Lax equation

represents the Hamiltonian system

If μ is a homogeneous operadic variable one can use the above Hamilton’s equations to obtain

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.*

Let A = {V, μ} be a binary algebra with operation We require that μ = μ(q, p) so that (μ,M) is an operadic Lax pair, i.e the operadic Lax equation

represents the Hamiltonian system of the harmonic oscillator.

Let x, y ∈ V . Assuming that |M| = 0 and |μ| = 1, one has

Therefore, one has

Let dim V = n. In a basis {e_{1}, . . . , e_{n}} of V , the structure constants of A are defined by

In particular,

By denoting it follows that

In particular, one has

**Lemma 5.1.*** Let dim V = 2 and Then the 2-dimensional binary operadic Lax equations read*

For the harmonic oscillator, define its auxiliary functions A_{±} and D_{±} by

Then one has the following

**Theorem 5.2.*** Let C _{β} ∈ R (β = 1, . . . , 8) be arbitrary real–valued parameters, ) and*

*Then (μ,M) is a 2-dimensional binary operadic Lax pair of the harmonic oscillator.*

**Idea of proof.** Denote

Define the matrix

Then it follows from Lemma 5.1 that the 2-dimensional binary operadic Lax equations read

Since the parameters C_{β} are arbitrary, the latter constraints imply Γ = 0. Thus one has to consider the following differential equations

By direct calculations [9] one can show that

Research was in part supported by the Estonian Science Foundation, Grant ETF 6912. More expanded version of the present paper is presented in [9].

- BabelonO, BernardD, Talon M (2003) Introduction to Classical Integrable Systems CambridgeUniv Press.
- GerstenhaberM (1963)Thecohomology structure of an associative ring. Annof Math78: 267–288.
- GerstenhaberM,GiaquintoA, SchackSD (1992) Algebras, bialgebras, quantum groups, and algebraic deformations. In“Deformation Theory and Quantum Groups with Applications to MathematicalPhysics”.GerstenhaberM,StasheffJ, EdsContemp Math 134: 51–92.
- KlugeL,PaalE (2001)On derivation deviations in an abstract pre-operad. Comm Algebra 29: 1609–1626.
- Kluge L, Paal E, Stasheff J (2000) Invitation to composition. Comm Algebra 28 1405–1422.
- LaxPD (1968) Integrals of nonlinear equations of evolution and solitary waves. Comm Pure Applied Math 21: 467–490.
- PaalE (2007) Invitation to operadic dynamics. J Gen Lie Theory Appl1: 57–63.
- Paal E, Virkepu J (2008) Note on operadic harmonic oscillator. Rep Math Phys61: 207–212.
- PaalE,VirkepuJ, 2D binary operadicLax representation for harmonic oscillator. PreprintarXiv:0803.0592 (math-ph)

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