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 \(F/\gamma _{k}\left( F\right) ^{^{\prime }}\).