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.
|Journal||Advances in Mathematics|
|State||Published - 16 Apr 2022|
- Azumaya algebra
- Graded algebras
- Graded division algebras
- Polynomial identities
- Verbally prime