Central Test Set and Central Test Rank of Free Nilpotent Lie Algebras of Rank 2

Source abstract

In this paper, we introduce and establish the concept of a central test set for free nilpotent Lie algebras. We prove that the central test rank of a free nilpotent Lie algebra Lc,2 of rank 2 and nilpotency class c = 3 equals 3. Furthermore, we show that a minimal central test set of L3,2 belongs to the lower-degree bracket structures and is explicitly given by the set 𝒯 = {[x,y], [[x,y],x], [x,y],y}. We also analyze higher nilpotency classes c ≥ 4 and delineate the degree conditions under which elements effectively eliminate non-central derivations.