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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-mail:**[email protected]

**Received date:** January 29, 2009; **Revised date:** March 03, 2009

**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 by using the operadic Lax equation. The operadic Lax representations for the harmonic oscilla- tor are constructed in 3-dimensional binary anti-commutative algebras. As an example, an operadic Lax representation for the harmonic oscillator in the Lie algebra sl(2) is constructed.

In Hamiltonian formalism, a mechanical system is described by the canonical variables q^{i}; pi and
their time evolution is prescribed by the Hamiltonian system

(1.1)

By a Lax representation [3, 1] of a mechanical system one means such a pair (L;M) of matrices (linear operators) L;M that the above Hamiltonian system may be represented as the Lax equation

(1.2)

Thus, from the algebraic point of view, mechanical systems can be described by linear operators, i.e by linear maps of a vector space V . As a generalization of this one can pose the fol- lowing question [4]: how to describe the time evolution of the linear operations (multiplications)

The algebraic operations (multiplications) can be seen as an example of the operadic variables [2]. If an operadic system depends on time one can speak about operadic dynamics [4]. The latter may be introduced by simple and natural analogy with the Hamiltonian dynamics. In particular, the time evolution of the operadic variables may be given by the operadic Lax equation. In [5, 6], a 2-dimensional binary operadic Lax representation for the harmonic oscillator was constructed. In the present paper we construct the operadic Lax representations for the harmonic oscillator in 3-dimensional binary anti-commutative algebras. As an example, an operadic Lax representation for the harmonic oscillator in the Lie algebra sl(2) is constructed.

Let K be a unital associative commutative ring, V be a unital K-module, and Hom For an operation 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, and Also, it is convenient to use the reduced degree Throughout this paper, we assume that

**Definition 2.1** (endomorphism operad [2]). For dene the partial compositions

The sequence equipped with the partial compositions i, is called the endomorphism operad of V .

**Definition 2.2** (total composition [2]). The total composition is dened
by

The pair Com is called the composition algebra of E.

**Definition 2.3 **(Gerstenhaber brackets [2]). The Gerstenhaber brackets [; ] are dened in
ComEV as a graded commutator by

The commutator algebra of Com is denoted as One can prove (e.g [2]) that is a graded Lie algebra. The Jacobi identity reads

Assume that and operations are dierentiable. Dynamics in operadic systems (operadic dynamics) may be introduced by

**Definition 3.1** (operadic Lax pair [4]). Allow a classical dynamical system to be described
by the Hamiltonian system (1.1). An operadic Lax pair is a pair (L;M) of linear operations such that the Hamiltonian system (1.1) may be represented as the operadic Lax
equation

Evidently, the degree constraints give rise to ordinary Lax equation (1.2) [3,1].

The Hamiltonian of the harmonic oscillator is

Thus, the Hamiltonian system of the harmonic oscillator reads

(3.1)

If μ is a linear algebraic operation we can use the above Hamilton equations to obtain

Therefore, we get the following linear partial dierential equation for (q; p):

(3.2)

By integrating (3.2) one can get sequences of operations called the operadic (Lax representations for) harmonic oscillator. Since the general solution of the partial dierential equations depends on arbitrary functions, these representations are not uniquely determined.

Let be a binary algebra with an operation For simplicity assume thatWe require that so that (μ,M) is an operadic Lax pair, i.e the Hamiltonian system (3.1) of the harmonic oscillator may be written as the operadic Lax equation

Let Assuming that

Therefore,

Let dim V = n. In a basis of V , the structure constants of A are dened by

In particular,

By denoting it follows that

**Lemma 5.1.** Matrices

give a 3-dimensional Lax representation for the harmonic oscillator.

**Lemma 5.2**. *Let dim V = 3 and M be dened as in Lemma 5.1. Then the 3-dimensional binary
operadic Lax equations read*

In what follows, consider only anti-commutative algebras. Then one has

**Corollary 5.3.** Let A be a 3-dimensional anti-commutative algebra, i.e

Then the operadic Lax equations for the harmonic oscillator read

For the harmonic oscillator, dene its auxiliary functions A± by

(5.1)

**Theorem 5.4. **Let be arbitrary real{valued parameters, such that

(5.2)

Let M be dened as in Lemma 5.1, and

(5.3)

Then (μ,M) is a 3-dimensional anti-commutative binary operadic Lax pair for the harmonic oscillator.

**Proof.** Denote

Dene the matrix

Then it follows from Corollary 5.3 that the 3-dimensional anti-commutative binary operadic Lax equations read

Since the parameters are arbitrary, not simultaneously zero, the latter constraints imply

Thus we have to consider the following dierential equations

We show that

First note that (I) immediately follows from the denition of

The proof of (II) can be found in [6] (Theorem 5.2 (I)).

It seems attractive to specify the coecients *C _{v}* in Theorem 5.4 by the initial conditions

The latter together with (5.1) yield the initial conditions for A±:

In what follows assume that and Other cases can be treated similarly. From (5.3) we get the following linear system:

(6.1)

One can easily check that the latter system can be uniquely solved with respect to 1; : : : ; 9):

Remark 6.1. Note that the parameters *Cv* have to satisfy condition (5.2) to get the operadic
Lax representation for the harmonic oscillaror.

**Definition 6.2.** If then the multiplication is called a dynamical deformation of (over
the harmonic oscillator). If then the multiplication is called dynamically rigid (over
the harmonic oscillator).

Example 7.1 (so(3)). As an example consider the Lie algebra so(3) with the structure equations

Thus, the nonzero structure constants are

Using the above initial conditions (6.1), we get

From this linear system it is easy to see that the only nontrivial constants are C9 = -C4 = 1. Replacing these constants into (5.3) we get

Thus we can see that the present selection of the parameters *Cv* (*v* = 1; : : : 9) via the struc-
ture constants of so(3) does not give rise to the operadic Lax representation for the harmonic
oscillator. Thus so(3) is dynamically rigid over the harmonic oscillator. This happens because
condition (5.2) is not satised.

**Example 7.2** (sl(2)). Finally consider the Lie algebra sl(2) with the structure equations

We can see that the nonzero structure constants are

System (6.1) reads

from which it follows that the only nontrivial constants are From (5.3) we get the operadic Lax system

The research was in part supported by the Estonian Science Foundation, Grant ETF 6912.

- O. Babelon, D. Bernard, and M. Talon. Introduction to Classical Integrable Systems. Cambridge Univ. Press, 2003.
- M. Gerstenhaber. The cohomology structure of an associative ring. Ann. of Math. 78 (1963), 267{288.
- P. D. Lax. Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Applied Math. 21 (1968), 467-490.
- E. Paal. Invitation to operadic dynamics. J. Gen. Lie Theory Appl. 1 (2007), 57-63.
- E. Paal and J. Virkepu. Note on operadic harmonic oscillator. Rep. Math. Phys. 61 (2008), 207-212.
- E. Paal and J. Virkepu. 2D binary operadic Lax representation for harmonic oscillator. Preprint arXiv:0803.0592.

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