MINIMAL NONNILPOTENT LEIBNIZ ALGEBRAS

Lindsey Bosko-Dunbar,J.T Hi̇rd,Jonathan D Dunbar,Kristen Stagg Rovira

doi:10.24330/ieja.768254

Abstract

We classify all nonnilpotent, solvable Leibniz algebras with the property that all proper subalgebras are nilpotent. This generalizes the work of [E. L. Stitzinger, Proc. Amer. Math. Soc., 28(1)(1971), 47-49] and [D. Towers, Linear Algebra Appl., 32(1980), 61-73] in Lie algebras. We show several examples which illustrate the differences between the Lie and Leibniz results. $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$

Highlights

Leibniz algebras were defined by Loday in [11]. They are a generalization of Lie algebras, removing the restriction that the product must be anti-commutative or that the squares of elements must be zero
One immediate consequence of this is that while the Lie algebra generated by a single element is necessarily onedimensional, the Leibniz algebra generated by a single element could be of any dimension
Minimal nonnilpotent Lie algebras were studied by Stitzinger in [12]

Summary

Introduction

Leibniz algebras were defined by Loday in [11]. They are a generalization of Lie algebras, removing the restriction that the product must be anti-commutative or that the squares of elements must be zero. An algebra L is called minimal nonnilpotent if L is nonnilpotent, solvable, and all proper subalgebras of L are nilpotent. Minimal nonnilpotent Lie algebras were studied by Stitzinger in [12]. A Leibniz algebra L is nilpotent if and only if every proper subalgebra of L is properly contained in its normalizer. Let M be a subalgebra of a Leibniz algebra L.

Results

Conclusion