We observe that matrices with noncommutative elements such that: 1) elements in the same column commute 2) commutators of the cross terms are equal: [Mij, Mkl] = [Mkj, Mil] (e.g. [M11, M22] = [M21, M12]), behave almost as well as matrices with commutative elements. There is natural definition of the determinant, Cramer’s inversion formula (Manin), we prove the Cayley-Hamilton theorem, the Newton identities, facts about block matrices, etc. Such matrices are the simplest examples in Manin’s theory of ”noncommutative symmetries” (1987-90). Second, we demonstrate that such matrices are ubiquitous in quantum integrability. They enter Talalaev’s hep-th/0404153 breakthrough formulas: det(∂z − LGaudin(z)), det(1 − e−∂zTY angian(z)) for the ”quantum spectral curve”, appear in separation of variables and Capelli identities. It is known that Manin’s matrices (and their q-analogs) include ”RTT=TTR ” and Cartier-Foata matrices. Third, we show that linear algebra theorems established for such matrices have various applications to quantum integrable systems and Lie algebras (Capelli identities, construction of new generators in Z(Ucrit ( gln)) and quantum conservation laws, Knizhnik-Zamolodchikov equation, etc.) We also propose a construction of quantum separated variables for the XXX-Heisenberg system
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