R-Matrix Quantization of the Elliptic Ruijsenaars–Schneider Model

It is shown that the classical L-operator algebra of the elliptic Ruijsenaars-Schneider model can be realized as a subalgebra of the algebra of functions on the cotangent bundle over the centrally extended current group in two dimensions. It is governed by two dynamical r and ?r-matrices satisfying a closed system of equations. The corresponding quantum R and ?R-matrices are found as solutions to quantum analogs of these equations. We present the quantum L-operator algebra and show that the system of equations on R and ?R arises as the compatibility condition for this algebra. It turns out that the R-matrix is twist-equivalent to the Felder elliptic R ^F -matrix with ?R playing the role of the twist. The simplest representation of the quantum L-operator algebra corresponding to the elliptic Ruijsenaars-Schneider model is obtained. The connection of the quantum L-operator algebra to the fundamental relation RLL=LLR with Belavin's elliptic R matrix is established. As a byproduct of our construction, we find a new N-parameter elliptic solution to the classical Yang-Baxter equation. Received: 17 March 1997 / Accepted: 8 July 1997     

On a Poisson–Lie Analogue of the Classical Dynamical Yang–Baxter Equation for Self-dual Lie Algebras

We derive a generalization of the classical dynamical Yang–Baxter equation (CDYBE) on a self-dual Lie algebra G by replacing the cotangent bundle T*G in a geometric interpretation of this equation by its Poisson–Lie (PL) analogue associated with a factorizable constant r-matrix on G. The resulting PL-CDYBE, with variables in the Lie group G equipped with the Semenov-Tian-Shansky Poisson bracket based on the constant r-matrix, coincides with an equation that appeared in an earlier study of PL symmetries in the WZNW model. In addition to its new group theoretic interpretation, we present a self-contained analysis of those solutions of the PL-CDYBE that were found in the WZNW context and characterize them by means of a uniqueness result under a certain analyticity assumption.     

Drinfeld–Sokolov Reduction for Difference Operators and Deformations of -Algebras¶I. The Case of Virasoro Algebra

We propose a q-difference version of the Drinfeld-Sokolov reduction scheme, which gives us q-deformations of the classical -algebras by reduction from Poisson-Lie loop groups. We consider in detail the case of SL ₂ . The nontrivial consistency conditions fix the choice of the classical r-matrix defining the Poisson-Lie structure on the loop group LSL ₂ , and this leads to a new elliptic classical r-matrix. The reduced Poisson algebra coincides with the deformation of the classical Virasoro algebra previously defined in [19]. We also consider a discrete analogue of this Poisson algebra. In the second part [31] the construction is generalized to the case of an arbitrary semisimple Lie algebra. Received: 20 April 1997 / Accepted: 22 July 1997     

Schlesinger Transformations and Quantum R-Matrices

 Schlesinger transformations are discrete monodromy preserving symmetry transformations of a meromorphic connection which shift by integers the eigenvalues of its residues. We study Schlesinger transformations for twisted -valued connections on the torus. A universal construction is presented which gives the elementary two-point transformations in terms of Belavin's elliptic quantum R-matrix. In particular, the role of the quantum deformation parameter is taken by the difference of the two poles whose residue eigenvalues are shifted. Elementary one-point transformations (acting on the residue eigenvalues at a single pole) are constructed in terms of the classical elliptic r-matrix. The action of these transformations on the τ-function of the system may completely be integrated and we obtain explicit expressions in terms of the parameters of the connection. In the limit of a rational R-matrix, our construction and the τ-quotients reduce to the classical results of Jimbo and Miwa in the complex plane. Received: 19 December 2001 / Accepted: 20 May 2002 Published online: 14 October 2002     

Modular Classes of Regular Twisted Poisson Structures on Lie Algebroids

We derive a formula for the modular class of a Lie algebroid with a regular twisted Poisson structure in terms of a canonical Lie algebroid representation of the image of the Poisson map. We use this formula to compute the modular classes of Lie algebras with a twisted triangular r-matrix. The special case of r-matrices associated to Frobenius Lie algebras is also studied.      

Quantum Dynamical R-Matrices and Quantum Frobenius Group

We propose an algebraic scheme for quantizing the rational Ruijsenaars-Schneider model in the R-matrix formalism. We introduce a special parametrization of the cotangent bundle over . In new variables the standard symplectic structure is described by a classical (Frobenius) r-matrix and by a new dynamical -matrix. Quantizing both of them we find the quantum L-operator algebra and construct its particular representation corresponding to the rational Ruijsenaars-Schneider system. Using the dual parametrization of the cotangent bundle we also derive the algebra for the L-operator of the hyperbolic Calogero-Moser system. Received: 24 January 1997 / Accepted: 17 March 1997     

Construction of the elliptic Gaudin system based on Lie algebra

Gaudin model is a very important integrable model in both quantum field theory and condensed matter physics. The integrability of Gaudin models is related to classical r-matrices of simple Lie algebras and semi-simple Lie algebra. Since most of the constructions of Gaudin models works concerned mainly on rational and trigonometric Gaudin algebras or just in a particular Lie algebra as an alternative to the matrix entry calculations often presented, in this paper we give our calculations in terms of a basis of the typical Lie algebra, A _n , B _n , C _n , D _n , and we calculate a classical r-matrix for the elliptic Gaudin system with spin.      

Global Gauge Anomalies in Two-Dimensional Bosonic Sigma Models

We revisit the gauging of rigid symmetries in two-dimensional bosonic sigma models with a Wess-Zumino term in the action. Such a term is related to a background closed 3-form H on the target space. More exactly, the sigma-model Feynman amplitudes of classical fields are associated to a bundle gerbe with connection of curvature H over the target space. Under conditions that were unraveled more than twenty years ago, the classical amplitudes may be coupled to the topologically trivial gauge fields of the symmetry group in a way which assures infinitesimal gauge invariance. We show that the resulting gauged Wess-Zumino amplitudes may, nevertheless, exhibit global gauge anomalies that we fully classify. The general results are illustrated on the example of the WZW and the coset models of conformal field theory. The latter are shown to be inconsistent in the presence of global anomalies. We introduce a notion of equivariant gerbes that allow an anomaly-free coupling of the Wess-Zumino amplitudes to all gauge fields, including the ones in non-trivial principal bundles. Obstructions to the existence of equivariant gerbes and their classification are discussed. The choice of different equivariant structures on the same bundle gerbe gives rise to a new type of discrete-torsion ambiguities in the gauged amplitudes. An explicit construction of gerbes equivariant with respect to the adjoint symmetries over compact simply connected simple Lie groups is given.     

Gaudin Model,KZ Equation and an Isomonodromic Problem on the Tours

This Letter presents a construction of isospectral problems on the torus. The construction starts from an SU(n) version of the XYZ Gaudin model recently studied by Kuroki and Takebe within the context of a twisted WZW model. In the classical limit, the quantum Hamiltonians of the generalized Gaudin model turn into classical Hamiltonians with a natural r-matrix structure. These Hamiltonians are used to build a nonautonomous multi-time Hamiltonian system, which is eventually shown to be an isomonodromic problem on the torus. This isomonodromic problem can also be reproduced from an elliptic analogue of the KZ equation for the twisted WZW model. Finally, a geometric interpretation of this isomonodromic problem is discussed in the language of a moduli space of meromorphic connections.     

Basic gerbe over non-simply connected compact groups

We present an explicit construction of the basic bundle gerbes with connection over all connected compact simple Lie groups. These are geometric objects that appear naturally in the Lagrangian approach to the WZW conformal field theories. Our work extends the recent construction of Meinrenken [The basic gerbe over a compact simple Lie group, L’Enseignement Mathematique, in press. arXiv:math. DG/0209194] restricted to the case of simply connected groups.     

Calogero-Moser Lax pairs with spectral parameter for general Lie algebras

We construct a Lax pair with spectral parameter for the elliptic Calogero-Moser Hamiltonian systems associated with each of the finite-dimensional Lie algebras, of the classical and of the exceptional type. When the spectral parameter equals one of the three half periods of the elliptic curve, our result for the classical Lie algebras reduces to one of the Lax pairs without spectral parameter that were known previously. These Calogero-Moser systems are invariant under the Weyl group of the associated untwisted affine Lie algebra. For non-simply laced Lie algebras, we introduce new integrable systems, naturally associated with twisted affine Lie algebras, and construct their Lax operators with spectral parameter (except in the case of G₂).     

Separation of Variables for the Ruijsenaars System

We construct a separation of variables for the classical n-particle Ruijsenaars system (the relativistic analog of the elliptic Calogero-Moser system). The separated coordinates appear as the poles of the properly normalised eigenvector (Baker-Akhiezer function) of the corresponding Lax matrix. Two different normalisations of the BA functions are analysed. The canonicity of the separated variables is verified with the use of the r-matrix technique. The explicit expressions for the generating function of the separating canonical transform are given in the simplest cases n=2 and n=3. Taking the nonrelativistic limit we also construct a separation of variables for the elliptic Calogero-Moser system. Received: 10 January 1997 / Accepted: 1 April 1997     

Operator-valued connections,Lie connections,and Gauge field theory

Noether's first theorem tells us that the global symmetry groupG _r of an action integral is a Lie group of point transformations that acts on the Cartesian product of the space-time manifold with the space of states and their derivatives. Gauge theory constructs are thus required for symmetry groups that act indiscriminately on the independent and dependent variables where the group structure can not necessarily be realized as a subgroup of the general linear group. Noting that the Lie algebra of a general symmetry groupG _r can be realized as a Lie algebrag _r of Lie derivatives on an appropriately structured manifold,G _r -covariant derivatives are introduced through study of connection 1-forms that take their values in the Lie algebrag _r of Lie derivatives (operator-valued connections). This leads to a general theory of operator-valued curvature 2-forms and to the important special class of Lie connections. The latter are naturally associated with the minimal replacement and minimal coupling constructs of gauge theory when the symmetry groupG _r is allowed to act locally. Lie connections give rise to the gauge fields that compensate for the local action ofG _r in a natural way. All governing field equations and their integrability conditions are derived for an arbitrary finite dimensional Lie group of symmetries. The case whereG _r contains the ten-parameter Poincaré group on a flat space-timeM ₄ is considered. The Lorentz structure ofM ₄ is shown to give a pseudo-Riemannian structure of signature 2 under the minimal replacement associated with the Lie connection of the local action of the Poincaré group. Field equations for the matter fields and the gauge fields are given for any system of matter fields whose action integral is invariant under the global action of the Poincaré group.     

Poisson Homology of r-Matrix Type Orbits I: Example of Computation

Abstract

In this paper we consider the Poisson algebraic structure associated with a classical r-matrix, i.e. with a solution of the modified classical Yang–Baxter equation. In Section 1 we recall the concept and basic facts of the r-matrix type Poisson orbits. Then we describe the r-matrix Poisson pencil (i.e the pair of compatible Poisson structures) of rank 1 or CP ⁿ-type orbits of SL(n, C). Here we calculate symplectic leaves and the integrable foliation associated with the pencil. We also describe the algebra of functions on CP ⁿ-type orbits. In Section 2 we calculate the Poisson homology of Drinfeld–Sklyanin Poisson brackets which belong to the r-matrix Poisson family.     

Some monodromy representations of generalized braid groups

A flat connection on the trivial bundle over the complement inC ⁿ of the complexification of the system of the reflecting hyperplanes of theB _n,D _n Coxeter groups is built from a simple Lie algebra and its representation. The corresponding monodromy representations of the generalized braid groupsXB _n,XD _n are computed in the simplest case.     

Dynamicalr-matrices on the affinizations of arbitrary self-dual Lie algebras

We associate a dynamicalr-matrix with any such subalgebraL of a finite dimensional self-dual Lie algebraA for which the scalar product ofA remains nondegenerate onL and there exists a nonempty open subsetĽ ⊂L so that the restriction of (ad λ)εEnd(A) toL is invertible ∨λεĽ. Thisr-matrix is also well-defined ifL is the grade zero subalgebra of an affine Lie algebraA obtained from a twisted loop algebra based on a finite dimensional self-dual Lie algebraG. Application of evaluation homomorphisms to the twisted loop algebras yields spectral parameter dependentG ⊗G-valued dynamicalr-matrices that are generalizations of Felder’s ellipticr-matrices. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. This work was supported in part by the Hungarian National Science Fund (OTKA) under T034170.