The notion of Homogeneous Varieties

Question

An algebraic variety is called homogeneous if there is an algebraic group acting transitively on it. I studied an article saying that The variety of free polynilpotent Lie algberas is homogeneous.(without details). May you please explain it in details? What can we say about free Leibniz algebras? in other words, can we say that Free Leibniz algebras are homogeneous?

Answer 1

In general, a variety of algebras over an infinite field is always homogeneous, see here, and Corollary $2$ here, where also a definition of "homogeneous" in this context is given (so there are different meanings of "homogeneous").