TY - JOUR

T1 - On regular G-gradings

AU - Aljadeff, Eli

AU - David, Ofir

N1 - Publisher Copyright:
© 2015 American Mathematical Society.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2015

Y1 - 2015

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.

UR - http://www.scopus.com/inward/record.url?scp=84922470990&partnerID=8YFLogxK

U2 - 10.1090/s0002-9947-2014-06200-4

DO - 10.1090/s0002-9947-2014-06200-4

M3 - 文章

AN - SCOPUS:84922470990

VL - 367

SP - 4207

EP - 4233

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 6

ER -