Representation theory of the reflection equation algebra I: A quantization of Sylvester's law of inertia

We prove a version of Sylvester's law of inertia for the Reflection Equation Algebra (=REA). We will only be concerned with the REA constructed from the R-matrix associated to the standard q-deformation of GL(N,C). For q positive, this particular REA comes equipped with a natural ∗-structure, by which it can be viewed as a q-deformation of the ∗-algebra of polynomial functions on the space of self-adjoint N-by-N-matrices. We will show that this REA satisfies a type I-condition, so that its irreducible representations can in principle be classified. Moreover, we will show that, up to the adjoint action of quantum GL(N,C), any irreducible representation of the REA is determined by its extended signature, which is a classical signature vector extended by a parameter in R/Z. It is this latter result that we see as a quantized version of Sylvester's law of inertia.