Specht property of varieties of graded Lie algebras

Let $U{T}_n(F)$ be the algebra of the $n\times n$ upper triangular matrices and denote $U{T}_n(F{)}^{(-)}$ the Lie algebra on the vector space of $U{T}_n(F)$ 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 ${\mathbb{Z}}_n$-graded identities of $U{T}_n(F{)}^{(-)}$ 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 $U{T}_n(F{)}^{(-)}$, is finitely based. Moreover we show that if $F$ is an infinite field of characteristic $p=2$ then the ${\mathbb{Z}}_3$-graded identities of $U{T}_3^{(-)}(F)$ do not satisfy the Specht property. More precisely, we construct explicitly an ideal of graded identities containing that of $U{T}_3^{(-)}(F)$, and which is not finitely generated as an ideal of graded identities.