Ruang Fase Tereduksi Grup Lie Aff (1)

Edi Kurniadi

doi:10.34312/jjom.v3i2.10653

Ruang Fase Tereduksi Grup Lie Aff (1)

Edi Kurniadi

Abstract

ABSTRAK

Dalam artikel ini dipelajari ruang fase tereduksi dari suatu grup Lie khususnya untuk grup Lie affine berdimensi 2. Tujuannya adalah untuk mengidentifikasi ruang fase tereduksi dari melalui orbit coadjoint buka di titik tertentu pada ruang dual dari aljabar Lie . Aksi dari grup Lie pada ruang dual menggunakan representasi coadjoint. Hasil yang diperoleh adalah ruang Fase tereduksi tiada lain adalah orbit coadjoint-nya yang buka di ruang dual . Selanjutnya, ditunjukkan pula bahwa grup Lie affine tepat mempunyai dua buah orbit coadjoint buka. Hasil yang diperoleh dalam penelitian ini dapat diperluas untuk kasus grup Lie affine berdimensi dan untuk kasus grup Lie lainnya.

ABSTRACT

In this paper, we study a reduced phase space for a Lie group, particularly for the 2-dimensional affine Lie group which is denoted by Aff (1). The work aims to identify the reduced phase space for Aff (1) by open coadjoint orbits at certain points in the dual space aff(1)* of the Lie algebra aff(1). The group action of Aff(1) on the dual space aff(1)* is considered using coadjoint representation. We obtained that the reduced phase space for the affine Lie group Aff(1) is nothing but its open coadjoint orbits. Furthermore, we show that the affine Lie group Aff (1) exactly has two open coadjoint orbits in aff(1)*. Our result can be generalized for the n(n+1) dimensional affine Lie group Aff(n) and for another Lie group.

Keywords

The Reduced Phase Space; Affine Lie Groups; Coadjoint Orbits; Coadjoint Representations

Full Text:

PDF [INDONESIA]

References

A. A. Kirillov, “Lectures on the Orbit Method, Graduate Studies in Mathematics,” Am. Math. Soc., vol. 64, 2004.

P. Arathoon, “A bijection between the adjoint and coadjoint orbit of a semidirect product,” Differ. Geom. Its Appl., vol. 62, pp. 267-282, 2019.

M. A. Alvarez and et al, “Contact and Frobenius solvable Lie algebras with abelian nilradical,” Comm. Algebra., vol. 46, pp. 4344–4354, 2018.

A. Diatta and B. Manga, “On properties of principal elements of frobenius lie algebras,” J. Lie Theory, vol. 24, no. 3, pp. 849–864, 2014.

A. Diatta, B. Manga, and A. Mbaye, “On systems of commuting matrices , Frobenius Lie algebras and Gerstenhaber’s Theorem,” no. February, pp. 0–12, 2020.

D. Burde, “Left-symmetric algebras, or pre-Lie algebras in geometry and physics,” Cent. Eur. J. Math., vol. 4, no. 3, pp. 323–357, 2015.

T. Xue, “Nilpotent coadjoint orbits in small characteristic,” J. Algebr., vol. 397, pp. 111-140, 2014.

D. Beltita and B. Cahen, “Contractions of Lie algebras with 2-dimensional generic coadjoint orbits,” Linear Algebra Appl., vol. 466, pp. 41--63, 2015.

M. Ben Halima, “Coadjoint Orbits of Certain Motion Groups and Their Coherent States,” vol. 9251, no. August, 2016.

Lang, Honglei and Z. Liu, “Coadjoint orbits of Lie groupoids,” J. Geom. Phsysics, vol. 129, pp. 217-232, 2018.

R. Abraham and J. E. Marsden, Foundations of Machanics, Second Edi. Canada: Addison-wesley Publishing Company, 1978.

Ayala,V, A. Da Silva, and M. Ferreira, “Affine and bilinear systems on Lie groups,” Syst. &Control Lett., vol. 117, pp. 23-29, 2018.

J. Souza, “Sufficient conditions for dispersiveness of invariant control affine system on the Heisenberg group,” Syst. &Control Lett., vol. 124, pp. 68-74, 2019.

W. Rump, “Affine stucture of decomposable solvable groups,” J. Algebr., vol. 556, pp. 725-749, 2020.

J. Stillwell, Naive Lie Theory. New York: Springer-Verlag, 2008.

DOI: https://doi.org/10.34312/jjom.v3i2.10653