Some irreducible components of the variety of complex $n+1$-dimensional Leibniz algebras

TL;DR

This paper identifies specific Leibniz algebras whose orbit closures form irreducible components in the variety of complex n-dimensional Leibniz algebras and computes their second cohomology groups.

Contribution

It introduces particular Leibniz algebras that define irreducible components and calculates their second cohomology groups, advancing the understanding of algebraic variety structures.

Findings

Identified Leibniz algebras forming irreducible components

Calculated bases of second cohomology groups for these algebras

Contributed to the classification of complex Leibniz algebra varieties

Abstract

In the present paper we indicate some Leibniz algebras whose closures of orbits under the natural action of $\GL_n$ form an irreducible component of the variety of complex $n$-dimensional Leibniz algebras. Moreover, for these algebras we calculate the bases of their second groups of cohomologies.