Specht property of varieties of graded Lie algebras

Daniela Martinez Correa; Plamen Koshlukov

paper

Specht property of varieties of graded Lie algebras

arXiv:2208.08550

Abstract

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 $Z_{n}$ -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 $Z_{3}$ -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.