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Algebraic properties of Manin matrices. I. (English) Zbl 1230.05043
Summary: We study a class of matrices with noncommutative entries, which were first considered by Yu. I. Manin in 1988 in relation with quantum group theory. They are defined as “noncommutative endomorphisms” of a polynomial algebra. More explicitly their defining conditions read: 1) elements in the same column commute; 2) commutators of the cross terms are equal: \([M_{ij}, M_{kl}] = [M_{kj}, M_{il}]\) (e.g. \([M_{11},M_{22}] = [M_{21},M_{12}]\)). The basic claim is that despite noncommutativity many theorems of linear algebra hold true for Manin matrices in a form identical to that of the commutative case. Moreover in some examples the converse is also true. The present paper gives a complete list and detailed proofs of algebraic properties of Manin matrices known up to the moment; many of them are new.
In particular we provide complete proofs that an inverse to a Manin matrix is again a Manin matrix and for the Schur formula for the determinant of a block matrix; we generalize the noncommutative Cauchy-Binet formulas discovered recently [S. Caracciolo, A. D. Sokal and A. Sportiello, Electron. J. Comb. 16, No. 1, Research Paper R103, 43 p. (2009; Zbl 1192.15001), arXiv:0809.3516], which includes the classical Capelli and related identities. We also discuss many other properties, such as the Cramer formula for the inverse matrix, the Cayley-Hamilton theorem, Newton and MacMahon-Wronski identities, Plücker relations, Sylvester’s theorem, the Lagrange-Desnanot-Lewis Caroll formula, the Weinstein-Aronszajn formula, some multiplicativity properties for the determinant, relations with quasideterminants, calculation of the determinant via Gauss decomposition, conjugation to the second normal (Frobenius) form, and so on and so forth. We refer to [A. V. Chervov and G. Falqui, J. Phys. A, Math. Theor. 41, No. 19, Article ID 194006, 28 p. (2008; Zbl 1151.81022), arXiv:0711.2236 and V. Rubtsov, A. V. Silantev and D. V. Talalaev, SIGMA, Symmetry Integrability Geom. Methods Appl. 5, Paper 110, 22 p., electronic only (2009; Zbl 1190.37079)] for some applications.
The bibliography contains 118 items.

MSC:
05A19 Combinatorial identities, bijective combinatorics
15B33 Matrices over special rings (quaternions, finite fields, etc.)
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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