On the algebraic variety of Hom-Lie algebras

TL;DR

This paper studies the algebraic geometric structure of the set of n-dimensional Hom-Lie algebras, describing their defining equations and geometric properties for various dimensions, and classifies certain subclasses related to quadratic operads.

Contribution

It characterizes the algebraic variety of Hom-Lie algebras for all dimensions and identifies the defining polynomial equations, including special cases for n=3 and n=4, and classifies P-algebra subclasses.

Findings

HLie(n) forms an algebraic subvariety of dimension n^2(n-1)/2

For n=3, HLie(n) coincides with the set of all n-dimensional Hom-Lie algebras

For n=4, HLie(n) is a hypersurface in K^{24}

Abstract

The set HLie(n) of the n-dimensional Hom-Lie algebras over an algebraically closed field of characteristic zero is provided with a structure of algebraic subvariety of the affine plane of dimension n^2(n-1)/2}. For n=3, these two sets coincide, for n=4 it is an hypersurface in K^{24}. For n>4, we describe the scheme of polynomial equations which define HLie(n). We determine also what are the classes of Hom-Lie algebras which are P-algebras where P is a binary quadratic operads.