2-capability and 2-nilpotent multiplier of finite dimensional nilpotent Lie algebras

Abstract

Abstract. In the present context, we investigate to obtain some more results about 2-nilpotent multiplier M (2) (L) of a finite dimensional nilpotent Lie algebra L. For instance, we characterize the structure of M (2) (H) when H is a Heisenberg Lie algebra. Moreover, we give some inequalities on dim M (2) (L) to reduce a well known upper bound on 2-nilpotent multiplier as much as possible. Finally, we show that H(m) is 2-capable if and only if m=1.

Surjektifitas Pemetaan Eksponensial untuk Grup Lie Heisenberg yang Diperumum

“…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]. Berbeda dengan penelitian-penelitian sebelumnya, pada penelitian ini dipelajari sifat aljabar Lie Heisenberg diperumum berdimensi 2n + 1.…”

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

“…In the context of a representation theory of a Lie group, we can see that the Heisenberg Lie group can be realized in the quaternionic stage (Balachandirin & et al, 2017) and modular representations over a Heisenberg algebras (Szechtman, 2014). Furthermore, the structure of generalization of the Heisenberg Lie algebra were studied in (Cantuba & Merciales, 2020), (Cantuba, 2019), (Souza, 2019), (Niroomand & Johari, 2018), (Nirooman & Parvizi, 2017), and (Liu & et al, 2012).…”

Characterizing nilpotent Lie algebras rely on the dimension of their $2$-nilpotent multipliers

“…Recall from [10], a Lie algebra L is said to be 2-capable if L ∼ = H/Z 2 (H) for a Lie algebra H. In the following corollary, we speciy which ones of Lie algebras with 0 ≤ s 2 (L) ≤ 6 are capable.…”

“…In [10], the second author showed that the dimension of the 2-nilpotent multiplier of an n-dimensional non-abelian nilpotent Lie algebra L with the derived subalgebra of dimension m is bounded by 1 3 (n − m) (n + 2m − 2)(n − m − 1) + 3(m − 1) + 3. Then dim M (2) (L) ≤ 1 3 n(n − 2)(n − 1) + 3 and so we have dim M (2) (L) = 1 3 n(n − 2)(n − 1) + 3 − s 2 (L) for some s 2 (L) ≥ 0.…”

“…Then dim M (2) (L) ≤ 1 3 n(n − 2)(n − 1) + 3 and so we have dim M (2) (L) = 1 3 n(n − 2)(n − 1) + 3 − s 2 (L) for some s 2 (L) ≥ 0. The structure of all non-abelian nilpotent Lie algebras is obtained when s 2 (L) = 0 in [10]. The current paper is devoted to obtain the structure of all nilpotent non-abelian Lie algebras L when 1 ≤ s 2 (L) ≤ 6.…”