Algebraic properties of Manin matrices 1

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.Comment: 80 page

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

We study a natural q-analogue of a class of matrices with noncommutative entries, which were first considered by Yu. I. Manin in 1988 in relation with quantum group theory, (called Manin Matrices in [5]) . These matrices we shall call q-Manin matrices(qMMs). They are defined, in the 2x2 case, by the relations M_21 M_12 = q M_12 M_21; M_22 M_12 = q M_12 M_22; [M_11;M_22] = 1/q M_21 M_12 - q M_12 M_21: They were already considered in the literature, especially in connection with the q-Mac Mahon master theorem [16], and the q-Sylvester identities [25]. The main aim of the present paper is to give a full list and detailed proofs of algebraic properties of qMMs known up to the moment and, in particular, to show that most of the basic theorems of linear algebras (e.g., Jacobi ratio theorems, Schhur complement, the Cayley-Hamilton theorem and so on and so forth) have a straightforward counterpart for q-Manin matrices. We also show how this classs of matrices ?ts within the theory of quasi-determninants of Gel'fand-Retakh and collaborators (see, e.g., [17]). In the last sections of the paper, we frame our definitions within the tensorial approach to non-commutative matrices of the Leningrad school, and we show how the notion of q-Manin matrix is related to theory of Quantum Integrable Systems.Comment: 62 pages, v.2 cosmetic changes, typos fixe

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

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 2×22×2 case by the following relations among their matrix entries: M21M12=qM12M21, M22M12 = qM12M22 [M11,M22]=q-1M21M12-qM12M21 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

Manin matrices and Talalaev's formula

We study special class of matrices with noncommutative entries and demonstrate their various applications in integrable systems theory. They appeared in Yu. Manin's works in 87-92 as linear homomorphisms between polynomial rings; more explicitly they 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. $[M_{11}, M_{22}]=[M_{21}, M_{12}]$). We claim that such matrices behave almost as well as matrices with commutative elements. Namely theorems of linear algebra (e.g., a natural definition of the determinant, the Cayley-Hamilton theorem, the Newton identities and so on and so forth) holds true for them. On the other hand, we remark that such matrices are somewhat ubiquitous in the theory of quantum integrability. For instance, Manin matrices (and their q-analogs) include matrices satisfying the Yang-Baxter relation "RTT=TTR" and the so--called Cartier-Foata matrices. Also, they enter Talalaev's hep-th/0404153 remarkable formulas: $det(\partial_z-L_{Gaudin}(z))$, det(1-e^{-\p}T_{Yangian}(z)) for the "quantum spectral curve", etc. We show that theorems of linear algebra, after being established for such matrices, have various applications to quantum integrable systems and Lie algebras, e.g in the construction of new generators in $Z(U(\hat{gl_n}))$ (and, in general, in the construction of quantum conservation laws), in the Knizhnik-Zamolodchikov equation, and in the problem of Wick ordering. We also discuss applications to the separation of variables problem, new Capelli identities and the Langlands correspondence.Comment: 40 pages, V2: exposition reorganized, some proofs added, misprints e.g. in Newton id-s fixed, normal ordering convention turned to standard one, refs. adde

Limits of Gaudin Systems: Classical and Quantum Cases

We consider the XXX homogeneous Gaudin system with N sites, both in classical and the quantum case. In particular we show that a suitable limiting procedure for letting the poles of its Lax matrix collide can be used to define new families of Liouville integrals (in the classical case) and new ''Gaudin'' algebras (in the quantum case). We will especially treat the case of total collisions, that gives rise to (a generalization of) the so called Bending flows of Kapovich and Millson. Some aspects of multi-Poisson geometry will be addressed (in the classical case). We will make use of properties of ''Manin matrices'' to provide explicit generators of the Gaudin Algebras in the quantum case

SU(3) Richardson-Gaudin models: three level systems

We present the exact solution of the Richardson-Gaudin models associated with the SU(3) Lie algebra. The derivation is based on a Gaudin algebra valid for any simple Lie algebra in the rational, trigonometric and hyperbolic cases. For the rational case additional cubic integrals of motion are obtained, whose number is added to that of the quadratic ones to match, as required from the integrability condition, the number of quantum degrees of freedom of the model. We discuss different SU(3) physical representations and elucidate the meaning of the parameters entering in the formalism. By considering a bosonic mapping limit of one of the SU(3) copies, we derive new integrable models for three level systems interacting with two bosons. These models include a generalized Tavis-Cummings model for three level atoms interacting with two modes of the quantized electric field.Comment: Revised version. To appear in Jour. Phys. A: Math. and Theo