Nilpotent Lie algebras of breadth type (0, 3)

Abstract

For a natural number m, a Lie algebra L over a field k is said to be of breadth type $(0,m)$ if the co-dimension of the centralizer of every non-central element is m. In this article, we classify finite dimensional nilpotent Lie algebras of breadth type $(0,3)$ over ${\mathbb{F}}_q$ of odd characteristic up to isomorphism. We also give a partial classification of the same over finite fields of even characteristic, $\mathbb{C}$ and $\mathbb{R}$. We also discuss 2-step nilpotent Camina Lie algebras.

KEYWORD:

• Breadth type
• Camina Lie algebras
• nilpotent Lie algebras

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 17B05
• 17B30

Disclosure Statement

There is no conflict of interest for this article.

Acknowledgments

We thank Pradeep Kumar Rai for his interest in this problem. We also thank the anonymous referee for his/her valuable comments.

Additional information

Funding

Rijubrata Kundu thanks IISER Mohali for the Institute Post-Doctoral Fellowship. Tushar Kanta Naik would like to acknowledge support from NBHM via grant 0204/3/2020/R&D-II/2475 during the early stage of this work. Anupam Singh is funded by SERB through CRG/2019/000271 for this research.