Noncommutative Poisson structure and invariants of matrices

We introduce a novel approach that employs techniques from noncommutative Poisson geometry to comprehend the algebra of invariants of two $n\times n$ matrices. We entirely solve the open problem of computing the algebra of invariants of two $4\times 4$ matrices. As an application, we derive the complete description of the invariant commuting variety of $4\times 4$ matrices and the fourth Calogero-Moser space.