The algebraic and geometric classification of nilpotent weakly associative and symmetric Leibniz algebras

Abstract

This paper is devoted to the complete algebraic and geometric classification of complex $4$-dimensional nilpotent weakly associative, complex $4$-dimensional symmetric Leibniz algebras, and complex $5$-dimensional nilpotent symmetric Leibniz algebras. In particular, we proved that the variety of complex $4$-dimensional symmetric Leibniz algebras has no Vergne--Grunewald--O'Halloran Property (there is an irreducible component formed by only nilpotent algebras), but on the other hand, it has Vergne Property (there are no rigid nilpotent algebras).

arXiv admin note: substantial text overlap with arXiv:2106.00336, arXiv:1912.02691, arXiv:1812.01442, arXiv:2109.00483

References in corpus (5)

• Non-associative central extensions of null-filiform associative algebras
• Poisson algebras and symmetric Leibniz bialgebra structures on oscillator Lie algebras
• There are no rigid filiform Lie algebras of low dimension
• On certain class of Leibniz algebras
• A diagram connecting Symmetric Leibniz algebras with Lie-admissible algebras

Cited by in corpus (5)

• The algebraic and geometric classification of transposed Poisson algebras
• Non-associative algebraic structures: classification and structure
• Transposed Poisson structures
• The algebraic and geometric classification of Zinbiel algebras
• Solvable Leibniz superalgebras whose nilradical has the characteristic sequence $(n-1,1\mid m)$ and nilindex $n+m$