TY - JOUR
T1 - On generic G-graded Azumaya algebras
AU - Aljadeff, Eli
AU - Karasik, Yakov
N1 - Publisher Copyright:
© 2022 Elsevier Inc.
PY - 2022/4/16
Y1 - 2022/4/16
N2 - 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.
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.
KW - Azumaya algebra
KW - Graded algebras
KW - Graded division algebras
KW - Polynomial identities
KW - Verbally prime
UR - http://www.scopus.com/inward/record.url?scp=85125123401&partnerID=8YFLogxK
U2 - 10.1016/j.aim.2022.108292
DO - 10.1016/j.aim.2022.108292
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AN - SCOPUS:85125123401
SN - 0001-8708
VL - 399
JO - Advances in Mathematics
JF - Advances in Mathematics
M1 - 108292
ER -