On generic G-graded Azumaya algebras

Eli Aljadeff; Yakov Karasik

doi:10.1016/j.aim.2022.108292

On generic G-graded Azumaya algebras

Eli Aljadeff, Yakov Karasik^*

^*Corresponding author for this work

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

Abstract

Let F be an algebraically closed field of characteristic zero and let G be a finite group. Consider G-graded simple algebras A which are finite dimensional and e-central over F, i.e. Z(A)_e:=Z(A)∩A_e=F. For any such algebra we construct a generic G-graded algebra U which is Azumaya in the following sense. (1) (Correspondence of ideals): There is one to one correspondence between the G-graded ideals of U and the ideals of the ring R, the e-center of U. (2) Artin-Procesi condition: U satisfies the G-graded identities of A and no nonzero G-graded homomorphic image of U satisfies properly more identities. (3) Generic: If B is a G-graded algebra over a field then it is a specialization of U along an ideal a∈spec(Z(U)_e) if and only if it is a G-graded form of A over its e-center. We apply this to characterize finite dimensional G-graded simple algebras over F that admit a G-graded division algebra form over their e-center.

Original language	English
Article number	108292
Journal	Advances in Mathematics
Volume	399
DOIs	https://doi.org/10.1016/j.aim.2022.108292
State	Published - 16 Apr 2022
Externally published	Yes

Keywords

Azumaya algebra
Graded algebras
Graded division algebras
Polynomial identities
Verbally prime

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10.1016/j.aim.2022.108292

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title = "On generic G-graded Azumaya algebras",

abstract = "Let F be an algebraically closed field of characteristic zero and let G be a finite group. Consider G-graded simple algebras A which are finite dimensional and e-central over F, i.e. Z(A)e:=Z(A)∩Ae=F. For any such algebra we construct a generic G-graded algebra U which is Azumaya in the following sense. (1) (Correspondence of ideals): There is one to one correspondence between the G-graded ideals of U and the ideals of the ring R, the e-center of U. (2) Artin-Procesi condition: U satisfies the G-graded identities of A and no nonzero G-graded homomorphic image of U satisfies properly more identities. (3) Generic: If B is a G-graded algebra over a field then it is a specialization of U along an ideal a∈spec(Z(U)e) if and only if it is a G-graded form of A over its e-center. We apply this to characterize finite dimensional G-graded simple algebras over F that admit a G-graded division algebra form over their e-center.",

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AB - Let F be an algebraically closed field of characteristic zero and let G be a finite group. Consider G-graded simple algebras A which are finite dimensional and e-central over F, i.e. Z(A)e:=Z(A)∩Ae=F. For any such algebra we construct a generic G-graded algebra U which is Azumaya in the following sense. (1) (Correspondence of ideals): There is one to one correspondence between the G-graded ideals of U and the ideals of the ring R, the e-center of U. (2) Artin-Procesi condition: U satisfies the G-graded identities of A and no nonzero G-graded homomorphic image of U satisfies properly more identities. (3) Generic: If B is a G-graded algebra over a field then it is a specialization of U along an ideal a∈spec(Z(U)e) if and only if it is a G-graded form of A over its e-center. We apply this to characterize finite dimensional G-graded simple algebras over F that admit a G-graded division algebra form over their e-center.

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KW - Graded algebras

KW - Graded division algebras

KW - Polynomial identities

KW - Verbally prime

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On generic G-graded Azumaya algebras

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