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Giambruno, Antonio; Souza, Manuela da Silva

doi:10.1016/j.jalgebra.2013.05.009

Cited by 13 publications

(6 citation statements)

References 6 publications

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“…Now we check that the identities (10) also hold. Since L i is the span of e i for each i ∈ Z, then The identities (10) hold since the homogeneous components of U 1 have dimension 1.…”

Section: Z-graded Identities Of U 1 In Characteristic Zeromentioning

confidence: 99%

“…We denote by I the T Z -ideal generated by the polynomials in ( 9) and (10). By Lemma 3.1, I ⊆ T Z (U 1 ) and, as char K = 0, the opposite inclusion will follow from the inclusions P g n ∩ T Z (U 1 ) ⊂ I, for every n-tuple g of elements in Z.…”

Section: Z-graded Identities Of U 1 In Characteristic Zeromentioning

confidence: 99%

“…Bases of the graded polynomial identities for these gradings were determined in [15]. Over a field of characteristic zero, it was proved in [10] that the algebra sl 2 (K) endowed with any one of these three non-trivial gradings satisfies the Specht property. In [11] the authors prove that, in each case, the corresponding variety of graded Lie algebras is a minimal variety of exponential growth.…”

Section: Introductionmentioning

confidence: 99%

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$\mathbb{Z}$-graded identities of the Lie algebras $U_1$

Fidelis¹,

Koshlukov²

2021

Preprint

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Let K be an infinite field of characteristic different from two and let U 1 be the Lie algebra of the derivations of the algebra of Laurent polynomials K[t, t −1 ]. The algebra U 1 admits a natural Z-grading. We provide a basis for the graded identities of U 1 and prove that they do not admit any finite basis. Moreover, we provide a basis for the identities of certain graded Lie algebras with a grading such that every homogeneous component has dimension ≤ 1, if a basis of the multilinear graded identities is known. As a consequence of this latter result we are able to provide a basis of the graded identities of the Lie algebra W 1 of the derivations of the polynomial ring K[t]. The Z-graded identities for W 1 , in characteristic 0, were described in [8]. As a consequence of our results, we give an alternative proof of the main result, Theorem 1, in [8], and generalize it to positive characteristic. We also describe a basis of the graded identities for the special linear Lie algebra slq(K) with the Pauli gradings where q is a prime number.

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“…Now we check that the identities (10) also hold. Since L i is the span of e i for each i ∈ Z, then The identities (10) hold since the homogeneous components of U 1 have dimension 1.…”

Section: Z-graded Identities Of U 1 In Characteristic Zeromentioning

confidence: 99%

Section: Z-graded Identities Of U 1 In Characteristic Zeromentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

$\mathbb{Z}$-graded identities of the Lie algebras $U_1$

Fidelis¹,

Koshlukov²

2021

Preprint

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show abstract

“…Bases of these graded identities were exhibited in [10] (it should be mentioned that the results in [10] hold for K infinite, char K = 2), see a streamlined and simplified version in [11]. Recently, it was proved in [5] that over a filed K of characteristic 0 the T-ideal of graded identities of sl 2 (K) has the Specht property. The growth of the variety of graded Lie algebras generated by sl 2 (K) was found in [6].…”

Section: Introductionmentioning

confidence: 99%

Z-graded identities of the Lie algebra W1

2015

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“…In [17] Koshlukov described the graded polynomial identities of sl 2 for the above gradings when the base field is infinite and of characteristic = 2. Recently Giambruno and Souza proved in [15] that the variety of graded Lie algebras generated by sl 2 is also Spechtian.…”

Section: Introductionmentioning

confidence: 99%