Graded central polynomials for the algebra \(M_n(K)\).
The description of the polynomial identities satisfied by the matrix algebra \(M_n(K)\) of order \(n\) over a field \(K\) is one of the most important and challenging goals in the PI theory. Such a description is known only when \(n=2\), and in the reviewer's opinion, the general case would remain open for quite a long time. Instead of considering such a ``hopeless'' problem one may study other classes of identities such as weak ones (almost as hopeless as the ordinary ones), identities with involution (the same). There are a few remarkable exceptions. A fundamental result obtained independently by Procesi and by Razmyslov describes the trace identities of \(M_n(K)\) in characteristic 0. Another exception is that of the graded polynomial identities. Recall that \(M_n(K)\) has a natural grading by the cyclic group of order \(n\). The corresponding graded identities were described by Vasilovsky (in characteristic 0), and by Azevedo (over any infinite field). The paper under review deals with the graded central polynomials for \(M_n(K)\) over an infinite field \(K\). The main result of the paper consists of producing finite generating sets of the graded T-spaces of the central polynomials for \(M_n(K)\) when the characteristic of \(K\) is 0 or a prime \(p>0\) such that \(p\) does not divide \(n\). As a further application such finite generating sets are found also when \(p=n\). The algebra \(M_n(K)\) admits a (finite) natural grading by the integers. The author describes finite generators of the corresponding graded central polynomials. In this case the description is uniform in all characteristics.
scientific article; zbMATH DE number 5566651