On the degenerations of solvable Leibniz algebras

TL;DR

This paper investigates the rigidity and degenerations of solvable Leibniz algebras, providing classifications, proving conjectures for low dimensions, and describing specific algebraic structures.

Contribution

It characterizes rigid solvable Leibniz algebras, describes four-dimensional cases, and confirms the Grunewald-O'Halloran conjecture for low-dimensional Lie and Leibniz algebras.

Findings

Rigidity results for solvable Leibniz algebras under certain nilradical conditions

Classification of four-dimensional solvable Leibniz algebras with three-dimensional rigid nilradicals

Validation of the Grunewald-O'Halloran conjecture for Lie algebras of dimension less than six and Leibniz algebras less than four

Abstract

The present paper is devoted to the description of rigid solvable Leibniz algebras. In particular, we prove that solvable Leibniz algebras under some conditions on the nilradical are rigid and we describe four-dimensional solvable Leibniz algebras with three-dimensional rigid nilradical. We show that the Grunewald-O'Halloran's conjecture "any $n$-dimensional nilpotent Lie algebra is a degeneration of some algebra of the same dimension" holds for Lie algebras of dimensions less than six and for Leibniz algebras of dimensions less than four. The algebra of level one, which is omitted in the 1991 Gorbatsevich's paper, is indicated.