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

Abstract

Let F be a free Lie algebra of rank n ≥ 2 and A be a free abelian Lie algebra of rank m ≥ 2. We prove that the test rank of the abelian product F ×A is m. Morever we compute the test rank of the algebra F/γ k (F) .

Abelian Product of Free Abelian and Free Lie Algebras

“…The Lie algebra A * ab Fn doesn't have test elements but in the free Lie algebra Fn there are many test elements. Interest in the test ranks of the abelian product A * ab Fn is explained in [4]. In [5,6] test sets and test ranks of solvable and metabelian products of groups were studied.…”