The Structure of The Group of Polynomial Matrices Unitary in The Indefinite Metric of Index 1

Abstract

We consider the group M of all polynomial matrices U(z) = U0 + U1*z + U2*z*z +...+Uk*z*...*z, k=0,1,... that satisfy equation U(z)*D*U(z)" = D with the diagonal n*n matrix D=diag{-1,1,1,...1}. Here n > 1, U(z)" = U0" + U1"*z + U2"*z*z + ..., and symbol A" for a constant matrix A denotes the Hermitiean conjugate of A. We show that the subgroup M0 of those U(z) in M, that are normalized by the condition U(0)=I, is the free product of certain groups. The matrices in each group-multiples are explicitly and uniquely parametrized so that every U=U(z) in M0 can be represented in the form U = G1 * G2 * ... * Gs with n*n polynomial matrix multiples G1, G2, ..., each of which belong to its group-multiple, and so that any two consecutive Gi and G(i+1) belong to two different group-multiples. The uniqueness of such parametrization for a given U includes the number of multiples s, their particular sequence G1,G2,... and the multiples themselves with their respective parametrization; all these items can be defined in only one way once the U is given.