Right and left solvable extensions of an associative Leibniz algebra

ABSTRACT

A sequence of nilpotent Leibniz algebras denoted by N_n,18 is introduced. Here n denotes the dimension of the algebra defined for n≥4; the first term in the sequence is ℛ₁₈ in the list of four-dimensional nilpotent Leibniz algebras introduced by Albeverio et al. [4]. Then all possible right and left solvable indecomposable extensions over the field ℝ are constructed so that N_n,18 serves as the nilradical of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program established to classify solvable Lie algebras using special properties rather than trying to extend one dimension at a time.

KEYWORDS:

• Derivation
• Leibniz algebra
• nil-independence
• nilpotency
• nilradical
• solvability

MATHEMATICS SUBJECT CLASSIFICATION:

• 17A30
• 17A32
• 17A36
• 17A60
• 17B30

Notes

¹ When we say a Leibniz algebra, we mean a right Leibniz algebra, unless the choice is understood from the context.

²When we work with the left Leibniz algebras, we interchange s and r in step (i) and (ii), the right multiplication operator to the left, and the right Leibniz identity to the left Leibniz identity in step (iii)