A quantization of Sylvester's law of inertia

Abstract: 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 $GL(N,\mathbb{C})$. 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 $\mathbb{R}/\mathbb{Z}$, and we will explain in what way this is a complete invariant of the orbits under the action by quantum $GL(N,\mathbb{C})$. This is part of a work in progress jointly with Stephen Moore.

quantum algebra

Audience: researchers in the topic

Quantum Groups Seminar [QGS]

Series comments: This seminar aims to bring together experts in the area of quantum groups. The seminar topics will cover the theory of quantum groups and related structures in a large sense: Hopf algebras, operator algebras, q-deformations, higher categories and related branches of noncommutative mathematics.

The zoom links will be distributed by mail, so please join the mailing list if you are interested in attending the seminar.