London Algebra Colloquium abstract
Verbally prime algebras: tensor products and PI equivalence
Plamen Kochloukov (Campinas, Brazil), 20th May 2010
Abstract
The verbally prime ideals of identities of associative algebras play a
prominent role in PI theory. These were described in concrete terms by
Kemer, under the assumption of characteristic 0 base field. He also showed
that the tensor product of any two verbally prime algebras satisfies the
same polynomial identities as another verbally prime algebra.
In this talk we shall discuss possible generalisations of this result to
the case of positive characteristic. It turns out one may obtain new (and
significantly simpler) proofs of the above mentioned result by means of
graded polynomial identities. Such an approach has an advantage that it is
(almost) characteristic-free. It will be shown that in positive
characteristic one does not have the PI equivalences from the
characteristic 0 case. Moreover we shall give a “measure” of the
distance among the ideals of identities for the algebras involved, in
terms of their Gelfand–Kirillov dimensions.