Résume | Sylvester's law of inertia states that two self-adjoint matrices A and B are related as A = X*BX for some invertible complex matrix X if and only if A and B have the same signature (N_+,N_-,N_0), i.e. the same number of positive, negative and zero eigenvalues. In this talk, we will discuss a quantized version of this law: we consider the reflection equation *-algebra (REA), which is a quantization of the *-algebra of polynomial functions on self-adjoint matrices, together with a natural adjoint action by quantum GL(N,C). We then show that to each irreducible bounded *-representation of the REA can be associated an extended signature (N_+,N_-,N_0,[r]) with [r] in R/Z, and we will explain in what way this is a complete invariant of the orbits under the action by quantum GL(N,C). This is part of a work in progress jointly with Stephen Moore. |