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

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.

On Properties of Five-dimensional Nonstandard Filiform Lie algebra

“…For 𝑘 = 5, then 𝒞 5 (𝔤) = [ 𝒞 4 (𝔤), 𝔤] = span{[𝑢, 𝑣]|𝑢 ∈ 𝔤 4 , 𝑣 ∈ 𝔤 } = span{𝟎} = {𝟎}. Thus, we find 𝑘 0 = 5 ∈ ℕ such that 𝒞 5 (𝔤) = {𝟎}.…”

Computational Thinking in Mathematics Learning: Systematic Literature Review

“…Teachers as educators are important to know and apply CT in learning aspects. CT is one of the necessary thinking skills in the 21st century (Kurniasi et al, 2022). From Figure 5, it can be seen that the method used the most is the descriptive method, whereas for development research and mixed method methods only a few have been carried out and published.…”

Surjektifitas Pemetaan Eksponensial untuk Grup Lie Heisenberg yang Diperumum

“…Penelitian tentang aljabar dan grup Lie Heisenberg telah banyak dilakukan misalnya tentang modul tak tereduksi baru untuk aljabar Lie Heisenberg baik dalam konteks aljabar Lie Hesienberg itu sendiri maupun penelitian-penelitian yang dikaitkan dengan struktur aljabar lainnya (lihat [1][2][3][4]). Salah satu hal yang menarik di sini bahwa aljabar Lie Heisenberg diperumum dapat menjadi pembentuk aljabar Lie Frobenius melalui jumlah langsung dengan split torus-nya [5], dan representasi grup Lie Heisenbergnya dapat direalisasikan dalam quaternionic stage [6]. Disisi lain penelitian tentang struktur aljabar Lie Heisenberg juga telah banyak diteliti khususnya yang berkaitan dengan polynomial Lie [7], struktur aljabar Lie Heisenberg mengenai dispersiveness of invariant sistem affine control [8], struktur dan pengali Schur aljabar Lie Heisenberg [4], dan 2-capability dan 2-nilpotent multiplier pada aljabar Lie Heisenberg [9].…”

“…Oleh karena itu, perhitungan eksponensial elemen-elemen di h n akan menggunakan Teorema 1 dan memanfaatkan formula Baker-Campbell-Hausdorff. Disisi lain, eksponensial dari matriks X dapat dihitung menggunakan formula pada persamaan (5).…”

“…Karena h n adalah aljabar Lie nilpoten 2 langkah, maka terdapat bilagan bulat positif n 0 = 3 sedemikian sehingga X n 0 = O. Dengan demikian, deret matriks pada persamaan (5) akan menjadi hingga dan eksponensial e X dapat ditentukan. Sebagai contoh eksponensial matriks R di bawah ini dapat dihitung dengan mudah menggunakan persamaan (5).…”