In this chapter we will define the category [email protected] of Cayley-Hamilton algebras of degree n. These are affine C-algebras A equipped with a trace map trA such that all trace identities holding in n×nmatrices also hold in A. Hence, we have to study trace identities and, closely related to them, necklace relations.This requires the description of the generic algebras∫
C〈x1, . . . , xm〉 = Tmn and ∮ n
C〈x1, . . . , xm〉 = Nmn
called the trace algebra of m generic n × n matrices, respectively the necklace algebra of m generic n × n matrices. For every A ∈ [email protected] there are epimorphisms Tmn -- A and Nmn -- trA(A) for some m. In chapter 2 we will reconstruct the Cayley-Hamilton algebra A (and its
central subalgebra trA(A)) as the ring of GLn-equivariant polynomial functions (resp. invariant polynomials) on the representation scheme repn A. Using the Reynolds operator in geometric invariant theory, it suffices to prove these results for the generic algebras mentioned above. An n-dimensional representation of the free algebra C〈x1, . . . , xm〉 is determined by the images of the generators xi in Mn(C), whence
repn C〈x1, . . . , xm〉 'Mn(C)⊕ . . .⊕Mn(C)︸ ︷︷ ︸ m
and the GLn-action on it is simultaneous conjugation. For this reason we have to understand the fundamental results on the invariant theory of m-tuples on n× n matrices, due to Claudio Procesi .