Correction of reference of Thm. 9.12 stating an equivalence of categories between modules over the rational Cherednik algebra and its spherical subalgebra
发表信息：
Algebra, etc.: in honor of Yu. I. Manin, Vol. I, 165-266, Progr. Math., 269, Birkhaeuser Boston, Inc., Boston, MA, 2009
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.
We study the properties of one-dimensional hypergeometric integral solutions of the q-difference ("quantum") analogue of the Knizhnik-Zamolodchikov-Bernard equations on tori. We show that they also obey a difference KZB heat equation in the modular parameter, give formulae for modular transformations, and prove a completeness result, by showing that the associated Fourier transform is invertible. These results are based on SL(3,Z) transformation properties parallel to those of elliptic gamma functions.
We construct a twisted version of the genus one universal Knizhnik-Zamolodchikov-Bernard (KZB) connection introduced by Calaque-Enriquez-Etingof, that we call the ellipsitomic KZB connection. This is a flat connection on a principal bundle over the moduli space of $Γ$-structured elliptic curves with marked points, where $Γ=\mathbb{Z}/M\mathbb{Z}\times\mathbb{Z}/N\mathbb{Z}$, and $M,N\geq1$ are two integers. It restricts to a flat connection on $Γ$-twisted configuration spaces of points on elliptic curves, which can be used to construct a filtered-formality isomorphism for some interesting subg…
Generalising work of Calaque-Enriquez-Etingof, we construct a universal KZB connection D_R for any finite (reduced, crystallographic) root system R. D_R is a flat connection on the regular locus of the elliptic configuration space associated to R, with values in a graded Lie algebra t_R with a presentation with relations in degrees 2, 3 and 4 which we determine explicitly. The connection D_R also extends to a flat connection over the moduli space of pointed elliptic curves. We prove that its monodromy induces an isomorphism between the Malcev Lie algebra of the elliptic pure braid group P_R co…
Let $(\otimes_{j=1}^nV_j)[0]$ be the zero weight subspace of a tensor product of finite-dimensional irreducible $\frak{sl}_2$-modules. The dynamical elliptic Bethe algebra is a commutative algebra of differential operators acting on $(\otimes_{j=1}^nV_j)[0]$-valued functions on the Cartan subalgebra of $\frak{sl}_2$. The algebra is generated by values of the coefficient $S_2(x)$ of a certain differential operator $D=(d/dx)^2+S_2(x)$, defined to V. Rubtsov, A. Silantyev, D. Talalaev in 2009. We express $S_2(x)$ in terms of the KZB operators introduced by G. Felder and C. Wieszerkowski in 1994. …
Cherednik attached to an affine Hecke algebra module a compatible system of difference equations, called quantum affine Knizhnik-Zamolodchikov (KZ) equations. In case of a principal series module we construct a basis of power series solutions of the quantum affine KZ equations. Relating the bases for different asymptotic sectors gives rise to a Weyl group cocycle, which we compute explicitly in terms of theta functions. For the spin representation of the affine Hecke algebra of type C the quantum affine KZ equations become the boundary qKZ equations associated to the Heisenberg spin-1/2 XXZ ch…
We construct a genus one analogue of the theory of associators and the Grothendieck-Teichmueller group. The analogue of the Galois action on the profinite braid groups is an action of the arithmetic fundamental group of a moduli space of elliptic curves on the profinite braid groups in genus one. This action factors through an explicit profinite group hat GT_ell, which admits an interpretation in terms of decorations of braided monoidal categories. We relate this group to its prounipotent group scheme version GT_ell(-). We construct a torsor over the latter group, the scheme of elliptic associ…
We present a direct construction of abstract generators for q-deformed W_N algebras. This procedure hinges upon a twisted trace formula for the elliptic algebra A_{q,p}(sl(N)_c) generalizing the previously known formulae for quantum groups.
Quantum universal enveloping algebras, quantum elliptic algebras and double (deformed) Yangians provide fundamental algebraic structures relevant for many integrable systems. They are described in the FRT formalism by R-matrices which are solutions of elliptic, trigonometric or rational type of the Yang--Baxter equation with spectral parameter or its generalization known as the Gervais--Neveu--Felder equation. While quantum groups and double Yangians appear as quasi-triangular Hopf algebras, this is no longer the case for elliptic algebras and the various deformations of Yangian type algebras.…
We construct a family of Poisson structures of hydrodynamic type on the loop space of $\mathbb{C} P^{n-1}$. This family is parametrized by the moduli space of elliptic curves or, in other words, by the modular parameter $τ$. This family can be lifted to a homogeneous Poisson structure on the loop space of $\mathbb{C}^n$ but in order to do that we need to upgrade the modular parameter $τ$ to an additional field $τ(x)$ with Poisson brackets $\{τ(x),τ(y)\}=0,~~\{τ(x),z_a(y)\}=2πi~ z_a(y)~δ^{\prime}(x-y)$ where $z_1,...,z_n$ are coordinates on $\mathbb{C}^n$. These homogeneous Poisson structures c…
By using twist construction, we obtain a quantum groupoid $\cald\ot_{q}\uqg$ for any simple Lie algebra $\frakg$. The underlying Hopf algebroid structure encodes all the information of the corresponding elliptic quantum group-the quasi-Hopf algebras as obtained by Fronsdal, Arnaudon et. al. and Jimbo et. al..
In this paper, we give a finite number of defining relations satisfied by a finite number of generators for the elliptic Lie algebras and superalgebras ${\frak g}_R$ with rank $\geq 2$. Here the $R$'s denote the reduced and non-reduced elliptic root systems with rank $\geq 2$. We also show that if ${\cal L}$ is an extended affine Lie algebra (EALA) whose non-isotropic roots form the $R$, then there exists a natural homomorphism ${\cal F}:{\frak g}_R \to{\cal L}$, which also give a universal central extension (UCE) surjective map from $[{\frak g}_R,{\frak g}_R]$ to the core of ${\cal L}$. (More…
A self-contained description of algebraic structures, obtained by combinations of various limit procedures applied to vertex and face sl(2) elliptic quantum affine algebras, is given. New double Yangians structures of dynamical type are in particular defined. Connections between these structures are established. A number of them take the form of twist-like actions. These are conjectured to be evaluations of universal twists.
The small elliptic quantum group $e_{τ,γ}(sl_N)$, introduced in the paper, is an elliptic dynamical analogue of the universal enveloping algebra $U(sl_n)$. We define highest weight modules, Verma modules and contragradient modules over $e_{τ,γ}(sl_N)$, the dynamical Shapovalov form for $e_{τ,γ}(sl_N)$ and the contravariant form for highest weight $e_{τ,γ}(sl_N)$-modules. We show that any finite-dimensional $sl_N$-module and any Verma module over $sl_N$ can be lifted to the corresponding $e_{τ,γ}(sl_N)$-module on the same vector space. For the elliptic quantum group $E_{τ,γ}(sl_N)$ we construct…
We present a systematic derivation of the abelianity conditions for the $q$-deformed $W$-algebras constructed from the elliptic quantum algebra $\mathcal{A}_{q,p}\big(\widehat{\mathfrak{gl}}(N)_{c}\big)$. We identify two sets of conditions on a given critical surface yielding abelianity lines in the moduli space ($p, q, c$). Each line is identified as an intersection of a countable number of critical surfaces obeying diophantine consistency conditions. The corresponding Poisson brackets structures are then computed for which some universal features are described.
The search for elliptic quantum groups leads to a modified quantum Yang-Baxter relation and to a special class of quasi-triangular quasi Hopf algebras. This paper calculates deformations of standard quantum groups (with or without spectral parameter) in the category of quasi-Hopf algebras. An earlier investigation of the deformations of quantum groups, in the category of Hopf algebras, showed that quantum groups are generically rigid: Hopf algebra deformations exist only under some restrictions on the parameters. In particular, affine Kac-Moody algebras are more rigid than their loop algebra q…
We construct a certain cross product of two copies of the braided dual $\tilde H$ of a quasitriangular Hopf algebra $H$, which we call the elliptic double $E_H$, and which we use to construct representations of the punctured elliptic braid group extending the well-known representations of the planar braid group attached to $H$. We show that the elliptic double is the universal source of such representations. We recover the representations of the punctured torus braid group obtained in arXiv:0805.2766, and hence construct a homomorphism to the Heisenberg double $D_H$, which is an isomorphism if…
The Yang-Baxter equation admits two classes of elliptic solutions, the vertex type and the face type. On the basis of these solutions, two types of elliptic quantum groups have been introduced (Foda et al., Felder). Fronsdal made a penetrating observation that both of them are quasi-Hopf algebras, obtained by twisting the standard quantum affine algebra U_q(g). In this paper we present an explicit formula for the twistors in the form of an infinite product of the universal R matrix of U_q(g). We also prove the shifted cocycle condition for the twistors, thereby completing Fronsdal's findings. …
