Recent research on polynomial identities

Question

I work in computational complexity, where I work on the problem of polynomial identity testing over arithmetic circuits. One particular case is when the variables over the polynomial ring don't commute. The Amitsur-Levitzky Theorem is a major negative result about the degree of the polynomial identities over the matrix rings.

A basic question in this area seems to be: "Can we find a basis of the polynomial identities for the algebra of $n \times n$ matrices, $n \geq 2 ?$ ". This seems to be resolved only for the case where $n=2$.

1. Are there any special/nice matrix algebras for which it might be able to find these basis elements? Or conversely, do we even have negative results that it might be hard to find basis elements for specific algebras?

2. Has there been any progress on the $n > 2$ case?

Any pointers to recent research in the area would be appreciated.

Answer 1

As far as I know little is known on the structure and characterization of polynomial identities. The known results are usually examples of new polynomial identities for matrix algebras, and also their relations to central polynomials.

For more information, I would recommend reading the book "Polynomial Identity Rings", by Drensky and Formanek - it is a really nice book with a lot of references! (plus, you may be able to search on google scholar for articles that cite this, and you may find more current research).

I hope this helps