In this paper we study properties of Lax and Transfer matrices associated with quantum integrable systems. Our point of view stems from the fact that their elements satisfy special commutation properties, considered by Yu. I. Manin some twenty years ago at the beginning of Quantum Group Theory. They are the commutation properties of matrix elements of linear homomorphisms between polynomial rings; more explicitly they read: 1) elements in the same column commute; 2) commutators of the cross terms are equal:The main aim of this paper is twofold: on the one hand we observe and prove that such matrices (which we call Manin matrices for short) 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) have a straightforward counterpart in the case of Manin matrices.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 remarkable formulas: det(∂ z − L Gaudin (z)), det(1 − e −∂z T Y angian (z)) for the "quantum spectral curve", and appear in separation of variables problem and Capelli identities.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 crit ( 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.

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: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, that is, Manin matrices are the most general class of matrices such that linear algebra holds true for them. 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 of Manin matrices in terms of matrix (Leningrad) notations; 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 [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. Finally several examples and open question are discussed. We refer to [CF07, RST08] for some applications in the realm of quantum integrable systems.

The higher Sugawara operators acting on the Verma modules over the affine Kac-Moody algebra at the critical level are related to the higher Hamiltonians of the Gaudin model due to work of Feigin, Frenkel and Reshetikhin. An explicit construction of the higher Hamiltonians in the case of gl n was given recently by Talalaev. We propose a new approach to these results from the viewpoint of the vertex algebra theory by proving directly the formulas for the higher order Sugawara operators. The eigenvalues of the operators in the Wakimoto modules of critical level are also calculated.

This paper is devoted to studying some properties of the Courant algebroids: we explain the so-called "conducting bundle construction" and use it to attach the Courant algebroid to Dixmier-Douady gerbe (following ideas of P. Severa). We remark that WZNW-Poisson condition of Klimcik and Strobl (math.SG/0104189) is the same as Dirac structure in some particular Courant algebroid. We propose the construction of the Lie algebroid on the loop space starting from the Lie algebroid on the manifold and conjecture that this construction applied to the Dirac structure above should give the Lie algebroid of symmetries in the WZNW-Poisson σ-model, we show that it is indeed true in the particular case of Poisson σ-model. 0 COURANT ALGEBROIDS 1 ContentsAt the end of this section we make a remark that WZNW-Poisson condition of Klimcik and Strobl [16] coincides with Dirac structure in the Courant algebroid twisted by the 3-form (this was discovered independently by P.Severa and A.Weinstein [27]).Section 5 is motivated by the papers [16,17] and we try to give more geometric understanding of some of their constructions. Consider the diagram:We will show that if A is an algebroid on Y then there pr * ev * A has a natural structure of algebroid, considering the T * M as an algebroid with the bracket constructed from Poisson bracket on M and considering its pullback pr * ev * T * M on the loop space LM we see that this algebroid is the same as algebroid considered in [17] which plays the role of symmetries in Poisson σ-model. Analogous algebroids were considered in [9,10]. We also hope that for a Courant algebroid on M one can naturally construct Lie algebroid on LM this construction should be agreed with the construction of line bundle on LM from gerbe on M and the transgression of characteristic class of gerbe to characteristic class of line bundle on LM.Remark on Notations. In this work will denote by O functions on our manifolds. Most of the considerations works for analytic or algebraic as well as for smooth functions. The basic situation in our paper that O are smooth functions. We are sorry, that it may cause some inconvenience for those readers, who get used to think about O only as about analytic or algebraic functions. The same words should be said about our notation T -it is tangent bundle, basic example is smooth tangent bundle for smooth manifold, but most considerations works well for the holomorphic tangent bundle as well.Remark on Further Developments. This paper was basically finished in summer 2001. Since that time there have been some further developments in the subject. One of the authors (P.B.) in [28] proposed the analogy between Courant algebroids and "vertex algebroids" i.e. algebroids that appeared in [22,23] and which gives rise to the chiral de Rham complex and also he gave a "coordinate free" construction and proved the uniqueness of the vertex algebroid. The paper [27] contains many important ideas related to the subject of present work.Acknowledgements. The first part of this work owes its existence to...

Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of n copies of the universal enveloping algebra U (g) of a semisimple Lie algebra g . This family is parameterized by collections of pairwise distinct complex numbers z1, . . . , zn . We obtain some new commutative subalgebras in U (g) ⊗n as limit cases of Gaudin subalgebras. These commutative subalgebras turn to be related to the hamiltonians of bending flows and to the Gelfand-Tsetlin bases. We use this to prove the simplicity of spectrum in the Gaudin model for some new cases.