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On solvable Lie algebras of small breadth

Borworn Khuhirun, Korkeat Korkeathikhun, Songpon Sriwongsa, Keng Wiboonton

Abstract

The concept of breadth has been used in the classification of p-groups and nilpotent Lie algebras. In this paper, we investigate this notion for finite-dimensional solvable Lie algebras. Our main focus is to characterize solvable Lie algebras of breadth less than or equal to 2. More importantly, we provide a complete classification of such Lie algebras that are pure and nonnilpotent over the complex numbers.

On solvable Lie algebras of small breadth

Abstract

The concept of breadth has been used in the classification of p-groups and nilpotent Lie algebras. In this paper, we investigate this notion for finite-dimensional solvable Lie algebras. Our main focus is to characterize solvable Lie algebras of breadth less than or equal to 2. More importantly, we provide a complete classification of such Lie algebras that are pure and nonnilpotent over the complex numbers.

Paper Structure

This paper contains 9 sections, 22 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Basic properties of breadth for Lie algebras
  3. Classification of solvable Lie algebras of breadth
  4. Solvable Lie algebra of type (S2)
  5. Solvable Lie algebra of type (S1)
  6. $\dim L^k=1$ for all $k\in\mathbb{Z}_{\geq2}$
  7. $\dim L^k=2$ for all $k\in\mathbb{Z}_{\geq2}$
  8. Concluding remarks
  9. Acknowledgements

Key Result

Theorem 2.1

KMS15$b(L)=1$ if and only if $\dim[L,L]=1$.

Theorems & Definitions (36)

  • Theorem 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Theorem 2.6
  • proof
  • ...and 26 more