A partial solution to a
problem of Procesi has recently been given by Formanek, Halpin, Li by determining
the Poincaré series of the ideal of two variable identities of M2(k). Two related
results are obtained in this article.
A weak identity of Mn(k) is a polynomial which vanishes identically on sln, the
subspace of Mn(k) of matrices of trace zero. We show that the Poincaré series of
the ideal of two variable weak identities of M2(k) is rational. In addition it is shown
that the ideal of identities of upper triangular 2 × 2 matrices in an arbitrary finite
number of variables has a rational Poincaré series. As an application we are able to
determine this ideal precisely.
