On nilpotency of outer pointwise inner actor of the lie algebra crossed modules

Authors

  • Farshid Saeedi Department of Mathematics, Mashhad Branch, Islamic Azad University
  • A. Allahyari Department of Mathematics, Mashhad Branch, Islamic Azad University

DOI:

https://doi.org/10.30495/jme.v14i0.962

Keywords:

Lie algebra crossed module, Nilpotency, Pointwise inner actor

Abstract

Let $\mathcal{L}$ be a Lie algebra crossed module and $\Act_{pi}(\mathcal{L})$ be a point wise inner Actor of $\mathcal{L}$. In this paper, we introduce lower and upper central series of $\mathcal{L}$ and show that if $\Act_{pi}(\frac{\mathcal{L}}{Z_j(\mathcal{L})}) / \Inn\Act(\frac{\mathcal{L}}{Z_j(\mathcal{L})}) $ is the nilpotent of class $k$, then $\Act_{pi}(\mathcal{L}) / \Inn\Act(\mathcal{L}) $ is the nilpotent of the maximum class $ j+k $. Moreover, if $ \dim(\mathcal{L}^i / (\mathcal{L}^i\cap Z_j(\mathcal{L})))\leqslant 1 $, then $ \Act_{pi}(\mathcal{L}) / \Inn\Act(\mathcal{L})$ is the nilpotent of the maximum class $ i+j-1$.

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Published

2019-04-10

Issue

Vol. 14

Section

Vol. 14, No. 1, (2020)

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