ABSTRACT

Tr(w! Tr(w2) • • • Tr(wk)) = T Y ^ ) TV(w2) • • • Tr(wk); in particular. Tr(Tr(^2) • • • Tr{wk)) = Tr(l) Tr(w2) • • • Tr(wk). Note that T is a commutative subalgebra of Tf{X}.

where the ^(jlv..,ju) G Q can be computed explicitly. (See also Example 12.21.)

Of course, Qk is a mixed trace polynomial that is homogeneous but not multilinear. Since the Hamilton-Cayley trace identity gk depends on fc, it is nontrivial, and our object here is to prove that every trace identity of •Mfc(Q) is a consequence of gk-This is a considerable improvement (and much easier to prove!) than Kemer's theorem, since the minimal identity S2k of Mfc(Q) does not generate id(Mn(Q)) as a T-ideal, cf. Exercise 3.6. However, it still remains a mystery as to how to prove that Mn is finitely based as a corollary of the Hellman-Procesi-Razmyslov theorem.