On regular G-gradings

Eli Aljadeff; Ofir David

doi:10.1090/s0002-9947-2014-06200-4

On regular G-gradings

Eli Aljadeff, Ofir David

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

Let A be an associative algebra over an algebraically closed field F of characteristic zero and let G be a finite abelian group. Regev and Seeman introduced the notion of a regular G-grading on A, namely a grading A =(formula present)that satisfies the following two conditions: (1) for every integer n ≥ 1 and every n-tuple (g1, g2, . . . , gn) ∈ Gⁿ, there are elements, a_i∈ A_{gi ,}i = 1, . . . ,n, such that Пⁿ₁a_i≠ = 0; (2) for every g, h ∈ G and for every a_g∈ A_g, b_h∈ A_h, we have a_gb_h= θ_g,hb_ha_gfor some nonzero scalar θ_g,h. Then later, Bahturin and Regev conjectured that if the grading on A is regular and minimal, then the order of the group G is an invariant of the algebra. In this article we prove the conjecture by showing that ord(G) coincides with an invariant of A which appears in PI theory, namely exp(A) (the exponent of A). Moreover, we extend the whole theory to (finite) nonabelian groups and show that the above result holds also in that case.

Original language	English
Pages (from-to)	4207-4233
Number of pages	27
Journal	Transactions of the American Mathematical Society
Volume	367
Issue number	6
DOIs	https://doi.org/10.1090/s0002-9947-2014-06200-4
State	Published - 2015
Externally published	Yes

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title = "On regular G-gradings",

abstract = "Let A be an associative algebra over an algebraically closed field F of characteristic zero and let G be a finite abelian group. Regev and Seeman introduced the notion of a regular G-grading on A, namely a grading A =(formula present)that satisfies the following two conditions: (1) for every integer n ≥ 1 and every n-tuple (g1, g2, . . . , gn) ∈ Gn, there are elements, ai∈ Agi ,i = 1, . . . ,n, such that Пn1ai≠ = 0; (2) for every g, h ∈ G and for every ag∈ Ag, bh∈ Ah, we have agbh= θg,hbhagfor some nonzero scalar θg,h. Then later, Bahturin and Regev conjectured that if the grading on A is regular and minimal, then the order of the group G is an invariant of the algebra. In this article we prove the conjecture by showing that ord(G) coincides with an invariant of A which appears in PI theory, namely exp(A) (the exponent of A). Moreover, we extend the whole theory to (finite) nonabelian groups and show that the above result holds also in that case.",

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N2 - Let A be an associative algebra over an algebraically closed field F of characteristic zero and let G be a finite abelian group. Regev and Seeman introduced the notion of a regular G-grading on A, namely a grading A =(formula present)that satisfies the following two conditions: (1) for every integer n ≥ 1 and every n-tuple (g1, g2, . . . , gn) ∈ Gn, there are elements, ai∈ Agi ,i = 1, . . . ,n, such that Пn1ai≠ = 0; (2) for every g, h ∈ G and for every ag∈ Ag, bh∈ Ah, we have agbh= θg,hbhagfor some nonzero scalar θg,h. Then later, Bahturin and Regev conjectured that if the grading on A is regular and minimal, then the order of the group G is an invariant of the algebra. In this article we prove the conjecture by showing that ord(G) coincides with an invariant of A which appears in PI theory, namely exp(A) (the exponent of A). Moreover, we extend the whole theory to (finite) nonabelian groups and show that the above result holds also in that case.

AB - Let A be an associative algebra over an algebraically closed field F of characteristic zero and let G be a finite abelian group. Regev and Seeman introduced the notion of a regular G-grading on A, namely a grading A =(formula present)that satisfies the following two conditions: (1) for every integer n ≥ 1 and every n-tuple (g1, g2, . . . , gn) ∈ Gn, there are elements, ai∈ Agi ,i = 1, . . . ,n, such that Пn1ai≠ = 0; (2) for every g, h ∈ G and for every ag∈ Ag, bh∈ Ah, we have agbh= θg,hbhagfor some nonzero scalar θg,h. Then later, Bahturin and Regev conjectured that if the grading on A is regular and minimal, then the order of the group G is an invariant of the algebra. In this article we prove the conjecture by showing that ord(G) coincides with an invariant of A which appears in PI theory, namely exp(A) (the exponent of A). Moreover, we extend the whole theory to (finite) nonabelian groups and show that the above result holds also in that case.

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On regular G-gradings

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