On Kirillov's lemma for nilpotent Lie algebras

Abstract

Abstract. We establish a sharpening of Kirillov's lemma on nilpotent Lie algebras with 1-dimensional center and use it to study the structure of 3-step nilpotent Lie algebras.

Classification of types A and A_+ from low dimensional standard and non-standard filiform Lie Algebras

“…Thus, 𝔷 1 (𝔤) is an abelian Lie algebra. Second, we recall the proof that 𝔷 2 (𝔤) is 2-step nilpotent Lie algebra in (Beltiţă & Beltiţă, 2015). By definition 6 in equation ( 5 We shall prove that 𝔷 2 (𝔤) 3 = {0}.…”

“…Furthermore, a second center of 𝔤 is given in the following form: Example 8 (Beltiţă & Beltiţă, 2015). Let 𝔤 be the four-dimensional standard filiform Lie algebra with basis 𝑆 = {𝑥 1 , … , 𝑥 4 }.…”

“…Example 11 (Beltiţă & Beltiţă, 2015). The Lie algebra 𝔥 = ⟨𝑥 1 , … , 𝑥 6 ⟩ with Lie brackets are [𝑥 6 , 𝑥 5 ] = 𝑥 4 , [𝑥 3 , 𝑥 2 ] = 𝑥 1 , [𝑥 6 , 𝑥 4 ] = 𝑥 1 is of type 𝐴 + .…”

“…Given an 𝑛-standard filiform Lie algebra, applying (Beltiţă & Beltiţă, 2015), we compute its first center and second center to determine its filiform Lie algebra whether type 𝐴 or/and type 𝐴 + . The method apply to case of five-dimensional non-standard filiform Lie algebras as well.…”

Struktur Simplektik pada Aljabar Lie Affine aff(2,R)

“…Aljabar Lie adalah ruang vektor atas lapangan F dengan bracket Lie yang memenuhi kondisi-kondisi pemetaan bilinear, skew-symmetric, dan identitas Jacobi [2]. Penelitian tentang aljabar Lie sudah banyak dilakukan, seperti representasi aljabar Lie [3][4][5][6], aljabar Lie nilpoten [7][8][9], dan penggunaan aljabar Lie untuk menyelesaikan model populasi biologis [10].…”