We introduce a novel approach that employs techniques from noncommutative Poisson geometry to comprehend the algebra of invariants of two n× n matrices. We entirely solve the open problem of computing the algebra of invariants of two 4 × 4 matrices. As an application, we derive the complete description of the invariant commuting variety of 4 × 4 matrices and the fourth Calogero-Moser space.
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Eshmatov et al. (2024) studied this question.
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