Abstract
Let F be an algebraically closed field of characteristic zero and let G be a finite group. We consider graded Verbally prime T-ideals in the free G-graded algebra. It turns out that equivalent definitions in the ordinary case (i.e. ungraded) extend to nonequivalent definitions in the graded case, namely verbally prime G-graded T-ideals and strongly verbally prime T-ideals. At first, following Kemer's ideas, we classify G-graded verbally prime T-ideals. The main bulk of the paper is devoted to the stronger notion. We classify G-graded strongly verbally prime T-ideals which are T-ideal of affine G-graded algebras or equivalently G-graded T-ideals that contain a Capelli polynomial. It turns out that these are precisely the T-ideal of G-graded identities of finite dimensional G-graded, central over F (i.e. Z(A)e=F) which admit a G-graded division algebra twisted form over a field k which contains F or equivalently over a field k which contains enough roots of unity (e.g. a primitive n-root of unity where n=ord(G)).
| Original language | English |
|---|---|
| Pages (from-to) | 142-175 |
| Number of pages | 34 |
| Journal | Advances in Mathematics |
| Volume | 332 |
| DOIs | |
| State | Published - 9 Jul 2018 |
| Externally published | Yes |
Keywords
- Graded algebras
- Graded division algebras
- Polynomial identities
- Verbally prime