Quantum currents realization of the elliptic quantum groups $E_{τ,η}({\frak{sl}}_{2})$
We review the construction by G. Felder and the author of the realization of the elliptic quantum groups of sl(2)-type by quantum currents.
q-alg↗
arXiv subjects
B. Enriquez
Publications and source records attributed to B. Enriquez.
Coinvariants for Yangian doubles and quantum KZ equations
We present a quantum version of the construction of the KZ system of equations as a flat connection on the spaces of coinvariants of representations of tensor products of Kac-Moody algebras. We consider here representations of a tensor product of Yangian doubles and compute the coinvariants of a deformation of the subalgebra generated by the regular functions of a rational curve with marked points. We observe that Drinfeld's quantum Casimir element can be viewed as a deformation of the zero-mode of the Sugawara tensor in the Yangian double. These ingredients serve to define a compatible system of difference equations, which we identify with the quantum KZ equations introduced by I. Frenkel and N. Reshetikhin.
q-alg↗
Dynamical $r$-matrices for Hitchin's systems on Schottky curves
We express Hitchin's systems on curves in Schottky parametrization, and construct dynamical $r$-matrices attached to them.
q-alg↗
Elliptic quantum groups $E_{τ,η}(sl_2)$ and quasi-Hopf algebras
We construct an algebra morphism from the elliptic quantum group $E_{τ,η}(sl_2)$ to a certain elliptic version of the ``quantum groups in higher genus'' studied by V. Rubtsov and the first author. This provides an embedding of $E_{τ,η}(sl_2)$ in an algebra ``with central extension''. In particular we construct $L^{\pm}$-operators obeying a dynamical version of the Reshetikhin--Semenov-Tian-Shansky relations. To do that, we construct the factorization of a certain twist of the latter algebra, that automatically satisfies the ``twisted cocycle condition'' of O. Babelon, D. Bernard and E. Billey, and therefore provides a solution of the dynamical Yang-Baxter equation.
q-alg↗
A construction of Hopf algebra cocycles for the double Yangian $DY(SL_{2})$
We construct a Hopf algebra cocycle in the Yangian double $DY(SL_{2})$, conjugating Drinfeld's coproduct to the usual one. To do that, we factorize the twist between two ``opposite'' versions of Drinfeld's coproduct, introduced in earlier work by V. Rubtsov and the first author, using the decomposition of the algebra in its negative and non-negative modes subalgebras.
q-alg↗
Separation of variables for Gaudin-Calogero systems
We construct an elliptic analogue of Sklyanin's separation of variables for the $sl(2)$ Gaudin system, using an adaptation of Drinfeld's Radon transformations.
q-alg↗
Hitchin systems, higher Gaudin operators and $r$-matrices
We adapt Hitchin's integrable systems to the case of a punctured curve. In the case of $\CC P^{1}$ 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 $GL_{n}$-bundles, and (for $n=2$) the corresponding quantum system.
alg-geom↗
Quantum groups in higher genus and Drinfeld's new realizations method ($sl_{2}$ case)
We define double (central and cocentral) extensions of Manin pairs introduced by Drinfeld, attached to curves and meromorphic differentials. We define ``infinite twistings'' of these pairs and quantize them in the $sl_{2}$ case, adapting Drinfeld's ``new realizations'' technique. We study finite dimensional representations of these algebras in level $0$, and some elliptic examples.
q-alg↗
Compatible Poisson-Lie structures on the loop group of $SL_{2}$
We define a 1-parameter family of $r$-matrices on the loop algebra of $sl_{2}$, defining compatible Poisson structures on the associated loop group, which degenerate into the rational and trigonometric structures, and study the Manin triples associated to them.
q-alg↗
Dispersionful analogues of Benney's equations and $N$-wave systems
We recall Krichever's construction of additional flows to Benney's hierarchy, attached to poles at finite distance of the Lax operator. Then we construct a ``dispersionful'' analogue of this hierarchy, in which the role of poles at finite distance is played by Miura fields. We connect this hierarchy with $N$-wave systems, and prove several facts about the latter (Lax representation, Chern-Simons-type Lagrangian, connection with Liouville equation, $τ$-functions).
solv-int↗
Soliton solutions of the classical lattice sine-Gordon system
We study the soliton-type solutions of the system introduced by B. Feigin and the author in in [EF]. We show that it reduces to a top-like system, and we study the behaviour of the solutions at the lattice infinity. We compute the scattering of the solitons and study some periodic solutions of the system.
hep-th↗
Integrals of motion of classical lattice sine-Gordon system
We compute the local integrals of motions of the classical limit of the lattice sine-Gordon system, using a geometrical interpretation of the local sine-Gordon variables. Using an analogous description of the screened local variables, we show that these integrals are in involution. We present some remarks on relations with the situation at roots of 1 and results on another latticisation (linked to the principal subalgebra of $\widehat{s\ell}_{2}$ rather than the homogeneous one). Finally, we analyse a module of ``screened semilocal variables'', on which the whole $\widehat{s\ell}_{2}$ acts.
hep-th↗
Quantum principal commutative subalgebra in the nilpotent part of $U_q\widehat{s\ell}_2$ and lattice KdV variables
We propose a quantum lattice version of Feigin and E. Frenkel's constructions, identifying the KdV differential polynomials with functions on a homogeneous space under the nilpotent part of $\widehat{s\ell}_2$. We construct an action of the nilpotent part $U_q\widehat n_+$ of $U_q\widehat{s\ell}_2$ on their lattice counterparts, and embed the lattice variables in a $U_q\widehat n_+$-module, coinduced from a quantum version of the principal commutative subalgebra, which is defined using the identification of $U_q\widehat n_+$ with its coordinate algebra.
hep-th↗
Nilpotent action on the KdV variables and 2-dimensional Drinfeld-Sokolov reduction
We note that a version ``with spectral parameter'' of the Drinfeld-Sokolov reduction gives a natural mapping from the KdV phase space to the group of loops with values in $\widehat N_{+}/A, \widehat N_{+}$~: affine nilpotent and $A$ principal commutative (or anisotropic Cartan) subgroup~; this mapping is connected to the conserved densities of the hierarchy. We compute the Feigin-Frenkel action of $\widehat n_{+}$ (defined in terms of screening operators) on the conserved densities, in the $sl_2$ case.
hep-th↗
Higher Gel'fand-Dikii structures
We apply the procedure of Magri and Weinstein to produce an infinity of compatible Poisson structures on a bihamiltonian manifold, to the case of the KdV phase space. The higher Gel'fand-Dikii structures thus obtained contain non local terms, which we express with the help of the r.h.s. of the KdV hierarchy. We also give a generating function for all these Poisson structues, in terms of the Baker-Akhiezer functions. Finally we describe the symplectic leaves of these Poisson structures.
hep-th↗