On Properties of  the (2n+1)-Dimensional Heisenberg Lie Algebra

Edi Kurniadi

doi:10.31764/jtam.v4i2.2339

On Properties of the (2n+1)-Dimensional Heisenberg Lie Algebra

Edi Kurniadi

Abstract

In the present paper, we study some properties of the Heisenberg Lie algebra of dimension . The main purpose of this research is to construct a real Frobenius Lie algebra from the Heisenberg Lie algebra of dimension . To achieve this, we exhibit how to compute the derivation of the Heisenberg Lie algebra by following Oom’s result. In this research, we use a literature review method to some related papers corresponding to a derivation of a Lie algebra, Frobenius Lie algebras, and Plancherel measure. Determining a conjecture of a real Frobenius Lie algebra is obtained. As the main result, we prove that conjecture. Namely, for the given the Heisenberg Lie algebra, there exists a commutative subalgebra of dimension one such that its semi direct sum is a real Frobenius Lie algebra of dimension . Futhermore, in the notion of the Lie group of the Heisenberg Lie algebra which is called the Heisenberg Lie group, we compute the generalized character of its group and we determine the Plancherel measure of the unitary dual of the Heisenberg Lie group. As our contributions, we complete some examples of Frobenius Lie algebras obtained from a nilpotent Lie algebra and we also give alternative computations to find the Plancherel measure of the Heisenberg Lie group.

Keywords

Heisenberg Lie algebra; Heisenberg Lie group; Frobenius Lie algebra; Generalized character; Unitary dual;Plancherel measure.

Full Text:

Download [PDF]

References

Alvarez, M. A., & et al. (2018). Contact and Frobenius solvable Lie algebras with abelian nilradical. Comm. Algebra, 46, 4344–4354.

Ayala, V., Kizil, E., & Tribuzy, I. D. A. (2012). On an algorithm for finding derivations of lie algebras. Proyecciones Journal of Mathematics, 31(1), 81–90. https://doi.org/10.4067/S0716-09172012000100008

Bagarello, F., & Russo, F. G. (2018). A description of pseudo-bosons in the terms of nilpotent Lie algebras. Journal of Geometry and Physics, 125, 1--11.

Balachandirin,Muraleetharan, & et al. (2017). A representation of Weyl_Heisenber Lie algebra in the quaternionic setting. Annal.of Physics, 385, 180--213.

Baraquin, I. (2017). Stocastics aspects of the unitary dual group,. Comptes Rendus Mathematique, 357, 180--185.

Bekkert,Victor, & et al. (2013). New irreducible modules for Heisenberg and affine Lie algebras. Journal of Algebra, 373, 284--298.

Cantuba, R. R. S. (2019). Lie polynomials in q-deformed Heisenberg algebras. Journal of Algebra, 522, 101--123.

Cantuba, R. R. S., & Merciales, Mark Anthony C. (2020). An extension of a q-deformed Heisenberg algebra and its Lie polynomials. Expositiones Mathematics.

Cebron,G, & Ulrich,M. (2016). Haar states and Levi processes on the unitary dual group,. Journal of Functional Analysis, 270, 2769--2811.

Corwin, L. ., & Greenleaf, F. . (1990). Representations of nilpotent Lie groups and their application. Part I. Basic theory and examples,. Cambridge: Cambridge University Press, Cambridge.

Diatta, A., & Manga, B. (2014). On properties of principal elements of frobenius lie algebras. J. Lie Theory, 24(3), 849–864.

Hilgert, J., & Neeb, K.-H. (2012). Structure and Geometry of Lie Groups. New York: Springer Monographs in Mathematics, Springer.

Kirillov, A. A. (2004). Lectures on the Orbit Method, Graduate Studies in Mathematics. American Mathematical Society,Providence, 64.

Kurniadi, E. (2020a). Construction of Real Frobenius Lie Algebras from Non Abelian Nilpotent Lie Algebras of Dimension ≤4. Submitted to Journal Ilmu Dasar UNEJ.

Kurniadi, E. (2020b). On Square-Integrable Representations of A Lie Group of 4-Dimensional Standard Filiform Lie Algebra. Cauchy:Jurnal Matematika Murni Dan Aplikasi.

Liu, D., & et al. (2012). Lie bialgebras structures on the twisted Heisenberg-Virasoro algebra. Journal of Algebra, 359, 35--48.

Lu,Rencai, & Zhao,Kaiming. (2015). Finite-dimensional simple modules over generalized Heisenberg algebras. Linear Algebra and Its Applications, 475, 276--291.

Nirooman,Peyman, & Parvizi, M. (2017). 2-capability and 2-nilpotent multiplier of finite dimensional nilpotent Lie algebras. Journal of Geometry and Physics, 121, 180--185.

Niroomand, P., & Johari, F. (2018). The structure, capability, and the Schur multiplier of the generelized Heisenberg Lie algebra. Journal of Algebra, 505, 482--489.

Ooms, A. I. (2009). Computing invariants and semi-invariants by means of Frobenius Lie algebras. J. Algebra, 321, 1293--1312. https://doi.org/10.1016/j.jalgebra.2008.10.026

Pham, D. N. (2016). G-Quasi-Frobenius Lie Algebras. Archivum Mathematicum, 52(4), 233–262. https://doi.org/10.5817/AM2016-4-233

Souza, J. . (2019). Sufficient conditions for dispersiveness of invariant control affine system on the Heisenberg group. System &Control Letters, 124, 68--74.

Stachura, P. (2013). On the quantum ax+b group. Journal of Geometry and Physics, 73, 125--149.

Szechtman, F. (2014). Modular representations of Heisenberg algebras. Linear Algebra and Its Applications, 457, 49--60.

Zeitlin,A.M. (2012). Unitary representations of a loop ax+b group, Wiener measure and Gamma -function. Journal of Functional Analysis, 263(3), 529--548.

DOI: https://doi.org/10.31764/jtam.v4i2.2339