A generalized Cayley–Hamilton theorem

Let FF be a solvable Lie subalgebra of the Lie algebra gln(C)gln(C) (=Cn×n=Cn×n as a vector space). Let fk(x1,x2,…,xp),fk(x1,x2,…,xp),(k=1,2,…,r),(k=1,2,…,r), be polynomials in the commuting variables x1,x2,…,xpx1,x2,…,xp with coefficients in CC. For n×nn×n matrices M1,M2,…,MrM1,M2,…,Mr, let F(x1,x2,…,xp)=∑k=1rMkfk(x1,x2,…,xp) and letδF(x1,x2,…,xp)=detF(x1,x2,…,xp).δF(x1,x2,…,xp)=detF(x1,x2,…,xp).In this paper, we prove that, for A1,A2,…,Ap,A1,A2,…,Ap,M1,M2,…,Mr∈FM1,M2,…,Mr∈F, if one value of the matrix-valued function F(A1,A2,…,Ap)F(A1,A2,…,Ap) (the value depends on the product order of the variables) is nilpotent, then, (a) all values of F(A1,A2,…,Ap)F(A1,A2,…,Ap) are nilpotent; (b) all values of δF(A1,A2,…,Ap)δF(A1,A2,…,Ap) (again depends on the product order of the variables) are nilpotent, and one value is 0. This generalizes the recent result in [7] and makes his result accurate. The main tool we use in this paper is the representation theory of solvable Lie algebras.