GRADED IDENTITIES AND PI EQUIVALENCE OF ALGEBRAS IN POSITIVE CHARACTERISTIC

ABSTRACT

The algebras M _{a, b} (E) ⊗ E and M _a+b (E) are PI equivalent over a field of characteristic 0 where E is the infinite-dimensional Grassmann algebra. This result is a part of the well-known tensor product theorem. It was first proved by Kemer in 1984–1987 (see Kemer 1991); other proofs of it were given by Regev (1990), and in several particular cases, by Di Vincenzo (1992), and by the authors (2004). Using graded polynomial identities, we obtain a new elementary proof of this fact and show that it fails for the T-ideals of the algebras M _{1, 1}(E) ⊗ E and M ₂(E) when the base field is infinite and of characteristic p > 2. The algebra M _{a, a} (E) ⊗ E satisfies certain graded identities that are not satisfied by M _2a (E). In another paper we proved that the algebras M _{1, 1}(E) and E ⊗ E are not PI equivalent in positive characteristic, while they do satisfy the same multilinear identities.

Key Words:

• Graded identities
• Polynomial identities
• T-prime T-ideal

2000 AMS MSC:

• 16R10
• 16R20
• 16S10
• 15A75

ACKNOWLEDGMENTS

We acknowledge the useful discussions with and suggestion given by O. M. Di Vincenzo. Thanks are due to the referee whose valuable remarks improved the paper significantly.

Sergio S. Azevedo is supported by postdoctoral grant from FAPESP, No. 02/11776-5, Marcello Fidelis is supported by PhD grant from CNPq, and Plamen Koshlukov is partially supported by CNPq.

Notes

^#Communicated by R. Parimala.