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Note on 2d binary operadic harmonic oscillator 1 | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Note on 2d binary operadic harmonic oscillator 1

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

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Abstract

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.

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) equationThe 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.

Operad

Let K be a unital associative commutative ring, and let Cn (n ∈ N) be unital K-modules. For f ∈ Cn, 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, Cf = Cn and οf = οn. Also, it is convenient to use the reduced degree |f| = n − 1. Throughout this paper, we assume that equation

Definition 2.1 (operad (e.g [2,3])). A linear (non-symmetric) operad with coefficients in K is a sequence C= {Cn}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

οi ∈ Hom(Cm ­ Cn,Cm+n−1), | οi | = 0

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

equation

(3) Unit I ∈ C1 exists such that

I ο0f = f = f οi 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 equation Hom equation Define the partial compositions for equation

equation

Then equation is an operad (with the unit equation ) 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.

Gerstenhaber brackets and operadic Lax pair

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

equation

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

equation

The commutator algebra of ComC is denoted as ComC= {C, [0≤, 0≤]}. One can prove that ComC is a graded Lie algebra. The Jacobi identity reads

equation

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

equation

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

equation

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

Operadic harmonic oscillator

Consider the Lax pair for the harmonic oscillator:

equation

Since the Hamiltonian is

equation

it is easy to check that the Lax equation

equation

represents the Hamiltonian system

equation

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

equation

Therefore, the linear partial differential equation for the operadic variable μ(q, p) reads

equation

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 equation We require that μ = μ(q, p) so that (μ,M) is an operadic Lax pair, i.e the operadic Lax equation

equation

represents the Hamiltonian system of the harmonic oscillator.

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

equation

Therefore, one has

equation

Let dim V = n. In a basis {e1, . . . , en} of V , the structure constants equation of A are defined by

equation

In particular,

equation

By denoting equation it follows that

equation

In particular, one has

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

equation

For the harmonic oscillator, define its auxiliary functions A± and D± by

equation

Then one has the following

Theorem 5.2. Let Cβ ∈ R (β = 1, . . . , 8) be arbitrary real–valued parameters,equation ) and

equation

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

Idea of proof. Denote

equation

Define the matrix

equation

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

equation

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

equation

By direct calculations [9] one can show that

equation

Acknowledgement

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

References

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