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Nilpotent Lie Algebras of breadth type $(0,3)$

Rijubrata Kundu, Tushar Kanta Naik, Anupam Singh

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 of dimension $m$. In this article, we classify finite dimensional nilpotent Lie algebras of breadth type $(0, 3)$ over $\mathbb F_q$ of odd characteristics 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.

Nilpotent Lie Algebras of breadth type $(0,3)$

Abstract

For a natural number , a Lie algebra over a field is said to be of breadth type if the co-dimension of the centralizer of every non-central element is of dimension . In this article, we classify finite dimensional nilpotent Lie algebras of breadth type over of odd characteristics up to isomorphism. We also give a partial classification of the same over finite fields of even characteristic, and . We also discuss -step nilpotent Camina Lie algebras.

Paper Structure

This paper contains 17 sections, 37 theorems, 36 equations, 1 table.

Table of Contents

  1. Introduction
  2. Notations
  3. Breadth type and Nilpotent Lie algebras
  4. Isoclinism of Lie algebras
  5. Classification of $(0,1)$ type
  6. Non-nilpotent Lie algebras over $\mathbb C$
  7. $p$-groups of exponent $p$ and nilpotent Lie algebras over $\mathbb{F}_p$
  8. $2$-step nilpotent Camina Lie algebras
  9. Rank $n$-subspaces of skew-symmetric matrices and Camina Lie algebras
  10. Classification of $(0, 2)$ type
  11. Nilpotent Lie algebras of breadth type $(0,3)$
  12. When $I$ is of dimension $1$
  13. When $I$ is of dimension $2$
  14. Proof of the main theorem
  15. $2$-step $4$-generated nilpotent Lie algebras over other fields
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $L$ be a finite dimensional nilpotent Lie algebra over finite field $\mathbb F_q$ of odd characteristic. Then, $L$ is of breadth type $(0,3)$ if and only if $L \cong \mathfrak{g}+ I$, where $I$ is some Abelian Lie algebra, and $\mathfrak{g}$ is one of the following:

Theorems & Definitions (65)

  • Theorem 1.1
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Proposition 2.4
  • Example 2.5: Heisenberg Lie algebra of dimension $2m+1$
  • Theorem 2.6
  • Theorem 2.7
  • Proposition 3.1
  • Proposition 3.2
  • ...and 55 more