Manin矩阵的代数性质1
Algebraic properties of Manin matrices 1
摘要：
本文研究了一类非交换元矩阵，这类矩阵是由俞敏洪首先考虑的。I. Manin在1988年与量子群理论的关系。它们被定义为多项式代数的“非交换自同态”。更明确地说，它们的定义条件是：1）同一列中的元素可交换; 2）交叉项的因子相等：$[M_{ij}, M_{kl}] = [M_{kj}, M_{il}]$（例如$[M_{11},M_{22}] = [M_{21},M_{12}]$）。其基本主张是，尽管存在非交换性，但线性代数的许多定理对Manin矩阵以与交换情形相同的形式成立。此外，在某些示例中，相反的情况也是如此。本文给出了一个完整的列表和详细的证明代数性质的马宁矩阵已知的时刻，其中许多是新的。特别是，我们提出的配方中的矩阵（列宁格勒）符号;提供完整的证明，逆M. M。又是一个M。对于分块矩阵行列式的Schur公式，我们推广了最近发现的非交换Cauchy-比内公式[arXiv：0809.3516]，其中包括经典的Capelli恒等式和相关恒等式.本文还讨论了逆矩阵的Cramer公式、Cayley-Hamilton定理、Newton恒等式和MacMahon-Wronski恒等式、Plucker关系、Sylvester定理、Lagrange-Desnanot-Lewis公式、Weinstein-Aronszajn公式、行列式的乘法性质、与拟行列式的关系、用Gauss分解计算行列式、第二个标准形式（Frobenius）的共轭，等等。对于某些应用，我们参考[arXiv：0711.2236]。
英文摘要：
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 present the formulation in terms of matrix (Leningrad) notations; provide complete proofs that an inverse to a M.m. is again a M.m. and for the Schur formula for the determinant of a block matrix; we generalize the noncommutative Cauchy-Binet formulas discovered recently [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, Plucker 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 [arXiv:0711.2236] for some applications.
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