Specht property of varieties of graded Lie algebras

Abstract

Let U TnpF q be the algebra of the n ˆn upper triangular matrices and denote U TnpF q p´q the Lie algebra on the vector space of U TnpF q 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 Zn-graded identities of U TnpF q p´q 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 TnpF q p´q , 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 p´q 3 pF q do not satisfy the Specht property. More precisely, we construct explicitly an ideal of graded identities containing that of U T p´q 3 pF q, and which is not finitely generated as an ideal of graded identities.