No abstract

We discuss various dualities, relating integrable systems and show that these dualities are explained in the framework of Hamiltonian and Poisson reductions. The dualities we study shed some light on the known integrable systems as well as allow to construct new ones, double elliptic among them. We also discuss applications to the (supersymmetric) gauge theories in various dimensions. 06/99

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.

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Hitchin systems, higher Gaudin operators and r-matrices B. Enriquez and V. Rubtsov Abstract. We adapt Hitchin's integrable systems to the case of a punctured curve. In the case of CP 1 and SL n -bundles, they are equivalent to systems studied by Garnier. The corresponding quantum systems were identified by B. Feigin, E. Frenkel and N. Reshetikhin with Gaudin systems. We give a formula for the higher Gaudin operators, using results of R. Goodman and N. Wallach on the center of the enveloping algebras of affine algebras at the critical level. Finally we construct a dynamical r-matrix for Hitchin systems for a punctured elliptic curve, and GL nbundles, and (for n = 2) the corresponding quantum system. Introduction. In [13], N. Hitchin introduced a class of integrable systems, attached to a complex curve X and a semisimple Lie group G. The problem of quantization of these systems was then connected by A. Beilinson and V. Drinfel'd to the Langlands program. This quantization makes use of differential operators on the moduli space of G-bundles on X, obtained from the action of the center of the local completion of the enveloping algebra of a Kac-Moody algebra, at the critical level.This program can also be carried out in the case of curves with marked points. In the particular case of the punctured CP 1 , the action of the center of the enveloping algebra was studied by B. Feigin, E. Frenkel and N. Reshetikhin in [6]; they obtained an integrable system whose first operators are identical to Gaudin's operators ([9]).In this paper, we consider the question of expressing the action of higher central elements. We first note, that the Adler-Kostant-Symes (AKS) scheme, which allows to write families of commuting operators ([2], [14], [21]), can be applied in the present situation, and then show that the higher Hamiltonians obtained in [6], coincide with those. So our problem turns out to be equivalent to expressing higher central elements in the enveloping algebras at critical level, a problem which was solved by several authors ([10], [12]). Here we show how the results of [10] can be used to derive a simple expression of higher Gaudin Hamiltonians.We then turn to the case of punctured elliptic curves. We show that the integrability of Hitchin's system can be deduced from an r-matrix argument. Here r-matrix relations contain additional terms, due to an invariance under the Cartan algebra action. The r-matrix depends on the moduli parameters, so it reminds dynamical r-matrices. In the case of one puncture, our L-operator and r-matrix seem connected with those considered respectively by I. Krichever and A. Gorsky and N. Nekrasov in [15] and [11], and H. Braden, T. Suzuki and E. Sklyanin [5], [19]. It is also analogous to the r-matrix appearing in the work of G. Felder and C. Wieczerkowski on the Knizhnik-Zamolodchikov-Bernard equation on elliptic curves ([7]). We give the form of the first Hamiltonians in this case; one of them contains a Weierstass potential, and so is analogous to the Calogero-Moser Hamiltonian. We comput...

We construct quasi-Hopf algebras quantizing double extensions of the Manin pairs of Drinfeld, associated to a curve with a meromorphic differential, and the Lie algebra sl 2 . This construction makes use of an analysis of the vertex relations for the quantum groups obtained in our earlier work, PBW-type results and computation of R-matrices for them; its key step is a factorization of the twist operator relating "conjugated" versions of these quantum groups.

We explain Sklyanin's separation of variables in geometrical terms and construct it for Hitchin and Mukai integrable systems. We construct Hilbert schemes of points on T * Σ for Σ = C, C * or elliptic curve, and on C 2 /Γ and show that their complex deformations are integrable systems of Calogero-Sutherland-Moser type. We present the hyperkähler quotient constructions for Hilbert schemes of points on cotangent bundles to the higher genus curves, utilizing the results of Hurtubise, Kronheimer and Nakajima. Finally we discuss the connections to physics of D-branes and string duality.01/99 IntroductionA way of solving a problem with many degrees of freedom is to reduce it to the problem with the smaller number of degrees of freedom. The solvable models allow to reduce the original system with N degrees of freedom to N systems with 1 degree of freedom which reduce to quadratures. This approach is called a separation of variables (SoV). Recently, E. Sklyanin came up with a "magic recipe" for the SoV in the large class of quantum integrable models with a Lax representation [1] [2]. The method reduces in the classical case to the technique of separation of variables using poles of the Baker-Akhiezer function, which goes back to the work [3], see also [4] for recent developments and more references.The basic strategy of this method is to look at the Lax eigen-vector ( which is the BakerAkhiezer function) Ψ(z, λ):L(z)Ψ(z, λ) = λ(z)Ψ(z, λ) (1.1) with some choice of normalization (this is the artistic part of the method). The poles z i of Ψ(z, λ) together with the eigenvalues λ i = λ(z i ) are the separated variables. In all the examples studied so far the most naive way of normalization leads to the canonically conjugate coordinates λ i , z i .The purpose of this paper is to explain the geometry behind the "magic recipe" in a broad class of examples, which include Hitchin systems [5], their deformations [6] and many-body systems considered as their degenerations [7][8]. We shall use the results of [9], [10], [11]. For a complex surface X let X [h] denote the Hilbert scheme of points on X of length h (if X is compact hyperkahler then so is X [h] [12]). Hitchin systemsHitchin systems can be thought of as a generalized many-body system. In fact, elliptic Calogero-Moser model as well as its various spin and some relativistic generalizations can be thought as of a particular degeneration of Hitchin system [7], [13], [8]. The integrable systemRecall the general Hitchin's setup [5]. One starts with the compact algebraic curve Σ of genus higher then one and a topologically trivial vector bundle V over it. Let G = SL N (C),The Hitchin system is the integrable system on the moduli space N of stable Higgs bundles. The point of N is the gauge equivalence class of a pair ( an operator 1 ∂ A =∂ +Ā, a holomorphic section φ of ad(V ) ⊗ ω 1 Σ ). The holomorphic structure on V is defined with the help of∂ A . The symplectic structure on N descends from the two-form Trδφ ∧ δĀ. The integrals of motion are the Hitchin's hamiltonians:Their t...

We generalize a recent observation [1] that the partition function of the 6vertex model with domain-wall boundary conditions can be obtained by computing the projections of the product of the total currents in the quantum affine algebra U q ( sl 2 ) in its current realization. A generalization is proved for the the elliptic current algebra [2,3]. The projections of the product of total currents are calculated explicitly and are represented as integral transforms of the product of the total currents. We prove that the kernel of this transform is proportional to the partition function of the SOS model with domain-wall boundary conditions.