The central polynomials for the Grassmann algebra

Brandão, Antônio Pereira; Koshlukov, Plamen; Krasilnikov, Alexei; Silva, Élida Alves da

doi:10.1007/s11856-010-0074-1

Cited by 15 publications

(40 citation statements)

References 19 publications

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“…If k ≥ m + n then Theorem 1.3 holds for the ideals T Our choice of the algebra A in the proof of Theorem 1.4 was made with a purpose to have the paper self-contained. 4. The tensor products of the form E ⊗ E r ⊗ · · · ⊗ E s were used to study the polynomial identities of Lie nilpotent associative algebras over a field of characteristic 0 by Drensky [8,Section 5].…”

Section: Proofs Of Theorems 13 and 14mentioning

confidence: 99%

“…Moreover, for some m and n (1) holds over an arbitrary ring R: for instance, T (3) T (3) ⊂ T (5) in R X for any R (see [5,Lemma 2.1]). However, in general Theorem 1.2 fails over Z and over a field of characteristic 3: it was shown in [7,14] that in this case T (3) T (2) T (4) and moreover, T (3) T (2) ℓ T (4) for all ℓ ≥ 1.…”

mentioning

confidence: 99%

“…Recently another proof of Theorem 1.2 has been published in [11]. 4. In [9] a pair (m, n) of positive integers was called null if for each algebra A (over a field F of characteristic 0)…”

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confidence: 99%

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Products of commutators in a Lie nilpotent associative algebra

Deryabina

Krasilnikov

2017

Journal of Algebra

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Let $F$ be a field and let $F \langle X \rangle$ be the free unital associative algebra over $F$ freely generated by an infinite countable set $X = \{x_1, x_2, \dots \}$. Define a left-normed commutator $[a_1, a_2, \dots, a_n]$ recursively by $[a_1, a_2] = a_1 a_2 - a_2 a_1$, $[a_1, \dots, a_{n-1}, a_n] = [[a_1, \dots, a_{n-1}], a_n]$ $(n \ge 3)$. For $n \ge 2$, let $T^{(n)}$ be the two-sided ideal in $F \langle X \rangle$ generated by all commutators $[a_1, a_2, \dots, a_n]$ ($a_i \in F \langle X \rangle)$. Let $F$ be a field of characteristic $0$. In 2008 Etingof, Kim and Ma conjectured that $T^{(m)} T^{(n)} \subset T^{(m+n -1)}$ if and only if $m$ or $n$ is odd. In 2010 Bapat and Jordan confirmed the "if" direction of the conjecture: if at least one of the numbers $m$, $n$ is odd then $T^{(m)} T^{(n)} \subset T^{(m + n -1)}.$ The aim of the present note is to confirm the "only if" direction of the conjecture. We prove that if $m = 2 m'$ and $n = 2 n'$ are even then $T^{(m)} T^{(n)} \nsubseteq T^{(m +n -1)}.$ Our result is valid over any field $F$.Comment: 8 pages, remarks and references added, typos fixe

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Section: Proofs Of Theorems 13 and 14mentioning

confidence: 99%

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confidence: 99%

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Products of commutators in a Lie nilpotent associative algebra

Deryabina

Krasilnikov

2017

Journal of Algebra

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“…Lemma 3. 8. If A has an elementary grading induced by an n-tuple of pairwise distinct elements of g then the orbit of every element in the set {1, 2, .…”

Section: Proposition 32 Letmentioning

confidence: 99%

Primeness property for graded central polynomials of verbally prime algebras

Diniz¹,

Fidelis²

2018

Journal of Pure and Applied Algebra

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Let F be an infinite field. The primeness property for central polynomials of Mn(F ) was established by A. Regev, i.e., if the product of two polynomials in distinct variables is central then each factor is also central. In this paper we consider the analogous property for Mn(F ) and determine, within the elementary gradings with commutative neutral component, the ones that satisfy this property, namely the crossed product gradings. Next we consider Mn(R), where R admits a regular grading, with a grading such that Mn(F ) is a homogeneous subalgebra and provide sufficient conditions -satisfied by Mn(E) with the trivial grading -to prove that Mn(R) has the primeness property if Mn(F ) does. We also prove that the algebras M a,b (E) satisfy this property for ordinary central polynomials. Hence we conclude that, over a field of characteristic zero, every verbally prime algebra has the primeness property.

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“…for any algebra A over any associative and commutative unital ring R (see [5, Lemma 2.1]). However, in general Theorem 1.2 fails over Z and over a field of characteristic 3: it was shown in [7,16] that in this case T (3) T (2) T (4) and moreover,…”

Section: Introductionmentioning

confidence: 99%

Products of several commutators in a Lie nilpotent associative algebra

Deryabina

Krasilnikov

2017

Int. J. Algebra Comput.

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Abstract. Let F be a field of characteristic = 2, 3 and let A be a unital associative F -algebra. Define a left-normed commutator [a 1 , a 2 , . .. . , m k be positive integers such that ℓ of them are odd and k − ℓ of them are even. LetThe aim of the present note is to show that, for any positive integers m 1 , . . . , m k , in general,Recently Dangovski has proved that if m 1 , . . . , m k are any positive integers then, in general,

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The central polynomials for the Grassmann algebra

Cited by 15 publications

References 19 publications

Products of commutators in a Lie nilpotent associative algebra

Products of commutators in a Lie nilpotent associative algebra

Primeness property for graded central polynomials of verbally prime algebras

Products of several commutators in a Lie nilpotent associative algebra

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