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Operadic representations of harmonic oscillator in some 3d algebras | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Operadic representations of harmonic oscillator in some 3d algebras

Eugen PAAL* and Juri 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-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

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 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.

Introduction

In Hamiltonian formalism, a mechanical system is described by the canonical variables qi; pi and their time evolution is prescribed by the Hamiltonian system

image (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

image(1.2)

Thus, from the algebraic point of view, mechanical systems can be described by linear operators, i.e by linear maps image 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) image

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.

Endomorphism operad and Gerstenhaber brackets

Let K be a unital associative commutative ring, V be a unital K-module, and imageHom image For an operationimage 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, image andimage Also, it is convenient to use the reduced degreeimage Throughout this paper, we assume that image

Definition 2.1 (endomorphism operad [2]). For image de ne the partial compositions

image

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

Definition 2.2 (total composition [2]). The total composition image is de ned by

image

The pair Com image is called the composition algebra of E.

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

image

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

image

Operadic Lax equation and harmonic oscillator

Assume that image and operations are di erentiable. 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 image such that the Hamiltonian system (1.1) may be represented as the operadic Lax equation

image

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

The Hamiltonian of the harmonic oscillator is

image

Thus, the Hamiltonian system of the harmonic oscillator reads

image (3.1)

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

image

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

image(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 di erential equations depends on arbitrary functions, these representations are not uniquely determined.

Evolution of binary algebras

Let image be a binary algebra with an operationimage For simplicity assume thatimageWe require that image 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

image

Let image Assuming thatimage

image

image

Therefore,

image

Let dim V = n. In a basisimage of V , the structure constantsimage of A are de ned by

image

In particular,

image

By denoting image it follows that

image

Main Theorem

Lemma 5.1. Matrices

image

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

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

image

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

image

Then the operadic Lax equations for the harmonic oscillator read

image

For the harmonic oscillator, de ne its auxiliary functions A± by

image(5.1)

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

image(5.2)

Let M be de ned as in Lemma 5.1, and

image (5.3)

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

Proof. Denote

image

De ne the matrix

image

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

image

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

Thus we have to consider the following di erential equations

image

We show that

image

First note that (I) immediately follows from the de nition of image

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

Initial conditions and dynamical deformations

It seems attractive to specify the coecients Cv in Theorem 5.4 by the initial conditions

image

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

image

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

image (6.1)

One can easily check that the latter system can be uniquely solved with respect to image1; : : : ; 9):

image

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. Ifimage then the multiplication  is called a dynamical deformation ofimage (over the harmonic oscillator). If image then the multiplication image is called dynamically rigid (over the harmonic oscillator).

Examples

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

image

Thus, the nonzero structure constants are

image

Using the above initial conditions (6.1), we get

image

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

image

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 satis ed.

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

image

We can see that the nonzero structure constants are

image

System (6.1) reads

image

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

image

Acknowledgement

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

References

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