Algebraic properties of Manin matrices II: q-analogues and integrable systems

作者：Chervov A; Falqui G; Rubtsov V^*; Silantyev A

来源：Advances in Applied Mathematics, 2014, 60: 25-89.

DOI：10.1016/j.aam.2014.06.001

• Non-commutative determinant
• Quasideterminant
• Manin matrix
• Jacobi ratio theorem
• Newton identities
• Cayley-Hamilton theorem
• Schur complement
• Dodgson condensation
• Lax matrix
• R-matrix

We study a natural q-analogue of a class of matrices with non-commutative entries, which were first considered by Yu.I. Manin in 1988 in relation with quantum group theory, (called Manin matrices in [5]). We call these q-analogues q-Manin matrices. These matrices are defined, in the 2 x 2 case by the following relations among their matrix entries: %26lt;br%26gt;M-21 M-12 = qM(12) M-21, M-22 M-12 = qM(12)M(22), %26lt;br%26gt;[M-11, M-22] = q(-1)M(21)M(12) - qM(12)M(21). %26lt;br%26gt;They were already considered in the literature, especially in connection with the q-MacMahon master theorem [10], and the q-Sylvester identities [22]. The main aim of the present paper is to give a full list and detailed proofs of the algebraic properties of q-Manin matrices known up to the moment and, in particular, to show that most of the basic theorems of linear algebras (e.g., Jacobi ratio theorems, Schur complement, the Cayley-Hamilton theorem and so on and so forth) have a straightforward counterpart for such a class of matrices. We also show how q-Manin matrices fit within the theory of quasideterminants of Gelfand-Retakh and collaborators (see, e.g., [11]). We frame our definitions within the tensorial approach to non-commutative matrices of the Leningrad school in the last sections. We finally discuss how the notion of q-Manin matrix is related to theory of Quantum Integrable Systems.

• 出版日期2014-9