Department of Mathematics, Tallinn University of Technology, Ehitajate tee 5, 19086 Tallinn, Estonia
Received date: January 29, 2009; Revised date: March 03, 2009
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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 qi; 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 that
We 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 Cv 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.
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