Surjektifitas Pemetaan Eksponensial untuk Grup Lie Heisenberg yang Diperumum

Edi Kurniadi; Putri Giza Maharani; Alit Kartiwa

doi:10.34312/jjom.v5i1.16721

Surjektifitas Pemetaan Eksponensial untuk Grup Lie Heisenberg yang Diperumum

Edi Kurniadi, Putri Giza Maharani, Alit Kartiwa

Abstract

The Heisenberg Lie Group is the most frequently used model for studying the representation theory of Lie groups. This Lie group is modular-noncompact and its Lie algebra is nilpotent. The elements of Heisenberg Lie group and algebra can be expressed in the form of matrices of size 3×3. Another specialty is also inherited by its three-dimensional Lie algebra and is called the Lie Heisenberg algebra. The Heisenberg Lie Group whose Lie Algebra is extended to the dimension 2n+1 is called the generalized Heisenberg Lie group and it is denoted by H whose Lie algebra is h_n. In this study, the surjectiveness of exponential mapping for H was studied with respect to h_n=⟨x ̅,y ̅,z ̅⟩ whose Lie bracket is given by [X_i,Y_i ]=Z. The purpose of this research is to prove the characterization of the Lie subgroup with respect to h_n. In this study, the results were obtained that if ⟨x ̅,y ̅ ⟩=:V⊆h_n a subspace and a set {e^(x_i ) e^(x_j ) ┤| x_i,x_j∈V }=:L⊆H then L=H and consequently Lie(L)≠V.

Keywords

Heisenberg Lie Group; Heisenberg Lie Algebra; Exponential Map; Surjectiveness

Full Text:

PDF

References

M. A. Alvarez, M. C. Rodr´ıguez-Vallarte, and G. Salgado, “Contact and Frobenius solvable Lie algebras with abelian nilradical,” Communications in Algebra, vol. 46, no. 10, pp.4344–4354, oct 2018, doi: 10.1080/00927872.2018.1439048.

F. Bagarello and F. G. Russo, “A description of pseudo-bosons in terms of nilpotent Lie algebras,” Journal of Geometry and Physics, vol.125, pp. 1–11, feb 2018, doi:10.1016/j.geomphys.2017.12.002.

R. R. S. Cantuba, “Lie polynomials in q-deformed Heisenberg algebras,” Journal of Algebra, vol. 522, pp. 101–123, mar 2019, doi:10.1016/j.jalgebra.2018.12.008.

P. Niroomand and F. Johari, “The structure, capability and the Schur multiplier of generalized Heisenberg Lie algebras,” Journal of Algebra, vol. 505, pp. 482–489, jul 2018, doi:10.1016/j.jalgebra.2018.03.014.

E. Kurniadi, “On Properties of the (2n+1)-Dimensional Heisenberg Lie Algebra,” JTAM (Jurnal Teori dan Aplikasi Matematika), vol. 4, no. 2, pp. 107–114, oct 2020, doi:10.31764/jtam.v4i2.2339.

B. Muraleetharan, K. Thirulogasanthar, and I. Sabadini, “A representation of Weyl–Heisenberg Lie algebra in the quaternionic setting,” Annals of Physics, vol. 385, pp. 180–213, oct 2017, doi: 10.1016/j.aop.2017.07.014.

R. R. S. Cantuba and M. A. C. Merciales, “An extension of a q-deformed Heisenberg algebra and its Lie polynomials,” Expositiones Mathematicae, vol. 39, no. 1, pp. 1–24, mar 2021, doi: 10.1016/j.exmath.2019.12.001.

J. A. Souza, “Sufficient conditions for dispersiveness of invariant control affine systems on the Heisenberg group,” Systems & Control Letters, vol. 124, pp. 68–74, feb 2019, doi:10.1016/j.sysconle.2018.12.004.

P. Niroomand and M. Parvizi, “2-capability and 2-nilpotent multiplier of finite dimensional nilpotent Lie algebras,” Journal of Geometry and Physics, vol. 121, pp. 180–185, nov 2017, doi:10.1016/j.geomphys.2017.07.003.

B. C. Hall, Lie Groups, Lie Algebras, and Representations, ser. Graduate Texts in Mathematics. Cham: Springer International Publishing, 2015, vol. 222, doi: 10.1007/978-3-319-13467-3.

L. Corwin and F. P. Greenleaf, “Representations of Nilpotent Lie Groups and their Applications,” in Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press, 2004, ch. 1.

J. Hilgert and K.-H. Neeb, Structure and Geometry of Lie Groups, ser. Springer Monographs in Mathematics. New York, NY: Springer New York, 2012, doi: 10.1007/978-0-387-84794-8.

V. V. Bavula, “The groups of automorphisms of the Lie algebras of formally analytic vector fields with constant divergence,” Comptes Rendus Mathematique, vol. 352, no. 2, pp. 85–88, feb 2014, doi:10.1016/j.crma.2013.12.001.

H. Henti, E. Kurniadi, and E. Carnia, “Levi Decomposition of Frobenius Lie Algebra of Dimension 6,” CAUCHY: Jurnal Matematika Murni dan Aplikasi, vol. 7, no. 3, pp. 394–400, oct 2022, doi: 10.18860/ca.v7i3.15656.

S. Khanal, R. R. Subedi, and G. Thompson, “Representations of nine-dimensional Levi decomposition Lie algebras,” Journal of Pure and Applied Algebra, vol. 224, no. 3, pp.1340–1363, mar 2020, doi:10.1016/j.jpaa.2019.07.020.

DOI: https://doi.org/10.34312/jjom.v5i1.16721