A Particular Test Element of a Free Solvable Lie Algebra of Rank Two

Test rank of an abelian product of a free Lie algebra and a free abelian Lie algebra

“…Examples of test elements of free Lie algebras of rank two were given by Mikhalev and Yu [10]. Other examples of test elements were considered by Mikhalev, Umirbaev and Yu [11], Temizyurek and Ekici [13] and Esmerligil [2].…”

“…It turns out that the test rank of a free polynilpotent Lie algebra is either equal to the rank of the Lie algebra or is one less than the rank of it. Interest in the test ranks of free soluble Lie algebras is explained in [13] and [15]. Temizyurek and Ekici [13] showed that a free soluble Lie algebra of rank 2 with solvability class 3 has test rank 1 and pointed out the particular test element for such algebras.…”

“…Interest in the test ranks of free soluble Lie algebras is explained in [13] and [15]. Temizyurek and Ekici [13] showed that a free soluble Lie algebra of rank 2 with solvability class 3 has test rank 1 and pointed out the particular test element for such algebras. The test rank for free soluble Lie algebras of solvability class greater than 3 is calculated by Timoshenko and Shevelin in [15] and it is shown that the test rank of a free soluble Lie algebra of rank n ≥ 2 is equal to n − 1.…”