The algebra of 2 × 2 upper triangular matrices as a commutative algebra: Gradings, graded polynomial identities and Specht property

Morais, Pedro; Souza, Manuela da Silva

doi:10.1016/j.jalgebra.2021.11.011

Cited by 6 publications

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“…The infinite dimensional Jordan algebra of a non-degenerate symmetric bilinear form also satisfies the Specht property in characteristic 0, this follows by combining results obtained by Vasilovsky [33] and by Koshlukov [16]. Recently, in [24] it was shown that for any grading, the variety of graded commutative algebras generated by pU T 2 , ˝q has the Specht property in characteristic 2.…”

Section: Introductionsupporting

confidence: 57%

Specht property of varieties of graded Lie algebras

Correa¹,

Koshlukov²

2022

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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.

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