Benjamin Enriquez

Elliptic Quantum Groups \(E_{\tau ,\eta } (\mathfrak{s}\mathfrak{l}_2 )\) and Quasi-Hopf Algebras

Abstract: We construct an algebra morphism from the elliptic quantum group

Quasi-hopf algebras associated withsl 2 and complex curves

E_{\tau ,\eta } (\mathfrak{s}\mathfrak{l}_2 )

Commuting families in skew fields and quantization of Beauville's fibration

to a certain elliptic version of the “quantum loop groups in higher genus” studied by V. Rubtsov and the first author. This provides an embedding of

Separation of variables for Gaudin–Calogero systems

E_{\tau ,\eta } (\mathfrak{s}\mathfrak{l}_2 )

Equivalence of Two Approaches to Integrable Hierarchies of KdV type

in an algebra “with central extension”. In particular we construct L±-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 quantum loop algebra, that automatically satisfies the “twisted cocycle equation” of O. Babelon, D. Bernard and E. Billey, and therefore provides a solution of the dynamical Yang–Baxter equation.

Examples of compact matrix pseudogroups arising from the twisting operation

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 algebrasl2. 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 ofR-matrices for them; its key step is a factorization of the twist operator relating “conjugated” versions of these quantum groups.

Integrals of motion of the classical lattice sine-Gordon system

We construct commuting families in fraction fields of symmetric powers of algebras. The classical limit of this construction gives Poisson commuting families associated with linear systems. In the case of a K3 surface S, they correspond to lagrangian fibrations introduced by Beauville. When S is the canonical cone of an algebraic curve C, we construct commuting families of differential operators on symmetric powers of C, quantizing the Beauville systems.

Quantum principal commutative subalgebra in the nilpotent part of\(U_q \widehat{sl}_2 \)and lattice KdV variables

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.

Rational forms for twistings of enveloping algebras of simple Lie algebras

The equivalence between the approaches of Drinfeld-Sokolov and Fei-gin-Frenkel to the mKdV hierarchies is established. A new derivation of the mKdV equations in the zero curvature form is given. Connection with the Baker-Akhiezer function and the tau-function is also discussed.

Complex parametrizedW-algebras: The gl case

We construct multiparameter compact matrix pseudogroups of all types and show their representation theories are the same as their classical analogs.