Identidade de Cayley-Hamilton para álgebras de matrizes.

Abstract

Over a field K of characteristic zero, we study in this dissertation the matrix algebras, Mn(K), from two points of view: rstly its trace identities - using the Invariant Theory as a basis - and, secondly, we provide the conditions for the realization of embeddings in this algebra, seeing it as a ring. Being more specic, we study the nature of the invariants of Mn(K), under the diagonal action of the general linear group, as well as the characterization of this ring as applications that depend of trace maps. Furthermore, we prove that all trace identities can be obtained by one called the Cayley-Hamilton polynomial of degree n, and also we prove that this ideal satisfies the Specht property. Lastly, using certain universal maps, we establish a condition for the existence of embeddings on the ring matrix of order n. With these results, we conclude that every nil algebra of bounded index n is a subring of Mn(C), for some commutative ring C.