Specht property of varieties of graded Lie algebras

Let UTn( F) be the algebra of the nxn upper triangular matrices and denote UTn(F)((-)) the Lie algebra on the vector space of UTn(F) with respect to the usual bracket (commutator), over an infinite field F. In this paper, we give a positive answer to the Specht property for the ideal of the Z(n)-graded identities of UTn(F)((-)) with the canonical grading when the characteristic p of F is 0 or is larger than n- 1. Namely we prove that every ideal of graded identities in the free graded Lie algebra that contains the graded identities of UTn(F)((-)), is finitely based. Moreover we show that if F is an infinite field of characteristic p = 2 then the Z(3)-graded identities of UT3(-)(F) do not satisfy the Specht property. More precisely, we construct explicitly an ideal of graded identities containing that of UT3(-)(F), and which is not finitely generated as an ideal of graded identities.