## Journal of Generalized Lie Theory and Applications

### Operadic representations of harmonic oscillator in some 3d algebras

#### 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 oscillator are constructed in 3-dimensional binary anti-commutative algebras. As an example, an operadic Lax representation for the harmonic oscillator in the Lie algebra $\mathfrak{sl}(2)$ is constructed.

#### Article information

Source
J. Gen. Lie Theory Appl., Volume 3, Number 1 (2009), Article ID S090104, 53-60.

Dates
First available in Project Euclid: 19 October 2011

https://projecteuclid.org/euclid.jglta/1319028475

Digital Object Identifier
doi:10.4303/jglta/S090104

Mathematical Reviews number (MathSciNet)
MR2486610

Zentralblatt MATH identifier
1163.18301

Subjects

#### Citation

PAAL, Eugen; VIRKEPU, Juri. Operadic representations of harmonic oscillator in some 3d algebras. J. Gen. Lie Theory Appl. 3 (2009), no. 1, Article ID S090104, 53--60. doi:10.4303/jglta/S090104. https://projecteuclid.org/euclid.jglta/1319028475

#### References

• 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.