Universal KZB equations I: the elliptic case

We define a universal version of the Knizhnik-Zamolodchikov-Bernard (KZB) connection in genus 1. This is a flat connection over a principal bundle on the moduli space of elliptic curves with marked points. It restricts to a flat connection on configuration spaces of points on elliptic curves, which can be used for proving the formality of the pure braid groups on genus 1 surfaces. We study the monodromy of this connection and show that it gives rise to a relation between the KZ associator and a generating series for iterated integrals of Eisenstein forms. We show that the universal KZB connection realizes as the usual KZB connection for simple Lie algebras, and that in the sl_n case this realization factors through the Cherednik algebras. This leads us to define a functor from the category of equivariant D-modules on sl_n to that of modules over the Cherednik algebra, and to compute the character of irreducible equivariant D-modules over sl_n which are supported on the nilpotent cone.Comment: Correction of reference of Thm. 9.12 stating an equivalence of categories between modules over the rational Cherednik algebra and its spherical subalgebr

On dynamical adjoint functor

We give an explicit formula relating the dynamical adjoint functor and dynamical twist over nonalbelian base to the invariant pairing on parabolic Verma modules. As an illustration, we give explicit $U(sl(n))$- and $U_\hbar(sl(n))$-invariant star product on projective spaces

A 2-categorical extension of Etingof–Kazhdan quantisation

Let k be a field of characteristic zero. Etingof and Kazhdan constructed a quantisation U_h(b) of any Lie bialgebra b over k, which depends on the choice of an associator Phi. They prove moreover that this quantisation is functorial in b. Remarkably, the quantum group U_h(b) is endowed with a Tannakian equivalence F_b from the braided tensor category of Drinfeld-Yetter modules over b, with deformed associativity constraints given by Phi, to that of Drinfeld-Yetter modules over U_h(b). In this paper, we prove that the equivalence F_b is functorial in b.Comment: Small revisions in Sections 2 and 6. An appendix added on the equivalence between admissible Drinfeld-Yetter modules over a QUE and modules over its quantum double. To appear in Selecta Math. 71 page

Parametrization of semi-dynamical quantum reflection algebra

We construct sets of structure matrices for the semi-dynamical reflection algebra, solving the Yang-Baxter type consistency equations extended by the action of an automorphism of the auxiliary space. These solutions are parametrized by dynamical conjugation matrices, Drinfel'd twist representations and quantum non-dynamical $R$-matrices. They yield factorized forms for the monodromy matrices.Comment: LaTeX, 24 pages. Misprints corrected, comments added in Conclusion on construction of Hamiltonian

Spin chains from dynamical quadratic algebras

We present a construction of integrable quantum spin chains where local spin-spin interactions are weighted by ``position''-dependent potential containing abelian non-local spin dependance. This construction applies to the previously defined three general quadratic reflection-type algebras: respectively non-dynamical, semidynamical, fully dynamical.Comment: 12 pages, no figures; v2: corrected formulas of the last sectio

Classical R-Matrices and the Feigin-Odesskii Algebra via Hamiltonian and Poisson Reductions

We present a formula for a classical $r$-matrix of an integrable system obtained by Hamiltonian reduction of some free field theories using pure gauge symmetries. The framework of the reduction is restricted only by the assumption that the respective gauge transformations are Lie group ones. Our formula is in terms of Dirac brackets, and some new observations on these brackets are made. We apply our method to derive a classical $r$-matrix for the elliptic Calogero-Moser system with spin starting from the Higgs bundle over an elliptic curve with marked points. In the paper we also derive a classical Feigin-Odesskii algebra by a Poisson reduction of some modification of the Higgs bundle over an elliptic curve. This allows us to include integrable lattice models in a Hitchin type construction.Comment: 27 pages LaTe