Related papers: A 2-categorical extension of Etingof-Kazhdan quant…

We discuss the proof of Kazhdan and Lusztig of the equivalence of the Drinfeld category D(g,h) of g-modules and the category of finite dimensional Uq(g)-modules, q=exp(\pi ih), for h\in C\Q*. Aiming at operator algebraists the result is…

For every semi-simple Lie algebra one can construct the Drinfeld-Jimbo algebra U. This algebra is a deformation Hopf algebra defined by generators and relations. To study the representation theory of U, Drinfeld used the KZ-equations to…

A braided tensor category $FM_{\kappa}$ of `factorizable D-modules' over configuration spaces is introduced, analogous to the category $FS_q$ of factorizable sheaves from q-alg/9604001. This category is equivalent to the category of finite…

Let ${\mathfrak g}$ be a complex semisimple Lie algebra, and $Y_h({\mathfrak g})$, $U_q(L{\mathfrak g})$ the corresponding Yangian and quantum loop algebra, with deformation parameters related by $q=\exp(\pi i h)$. When $h$ is not a…

This paper is a continuation of the series of papers "Quantization of Lie bialgebras (QLB) I-V". We show that the image of a Kac-Moody Lie bialgebra with the standard quasitriangular structure under the quantization functor defined in…

Working over a field $k$ of characteristic zero, the category of analytic contravariant functors on the category of finitely-generated free groups is shown to be equivalent to the category of representations of the $k$-linear category…

For a Hopf algebra B, we endow the Heisenberg double H(B^*) with the structure of a module algebra over the Drinfeld double D(B). Based on this property, we propose that H(B^*) is to be the counterpart of the algebra of fields on the…

We propose a variant to the Etingof-Kazhdan construction of quantization functors. We construct the twistor J_\Phi associated to an associator \Phi using cohomological techniques. We then introduce a criterion ensuring that the ``left Hopf…

In this paper, we construct a functorial quantization of (co)Poisson Hopf algebras within a broad categorical framework. We further introduce categories naturally associated with (co)Poisson Hopf algebras, namely Drinfeld-Yetter modules.…

We settle several questions about the theory of universal deformation quantization of Lie bialgebras by giving their complete classification up to homotopy equivalence. An important new technical ingredient introduced in this paper is an…

We study the behavior of the Etingof-Kazhdan quantization functors under the natural duality operations of Lie bialgebras and Hopf algebras. In particular, we prove that these functors are "compatible with duality", i.e., they commute with…

A positive level Kazhdan-Lusztig functor is defined using Arkhipov-Gaitsgory duality for affine Lie algebras. The functor sends objects in the DG category of G(O)-equivariant positive level affine Lie algebra modules to objects in the DG…

What are the fiber functors on small additive monoidal categories C which are not abelian? We give an answer which leads to a new Tannaka duality theorem for bialgebroids generalizing earlier results by Phung Ho Hai. The construction…

We show that any coboundary Lie bialgebra can be quantized. For this, we prove that: (a) Etingof-Kazhdan quantization functors are compatible with Lie bialgebra twists, and (b) if such a quantization functor corresponds to an even…

This article is a sequel to hep-th/9411050, q-alg/9412017. In Chapter 1 we associate with every Cartan matrix of finite type and a non-zero complex number $\zeta$ an abelian artinian category $\FS$. We call its objects {\em finite…

We construct a fully faithful functor from the category C_F of finite-dimensional representations of Felder's (dynamical) elliptic quantum group E_{tau,gamma}(gl(n)) to a cretain category D_B of (infinite-dimensional) representations of…

Let $(g,\delta_\hbar)$ be a Lie bialgebra. Let $(U_\hbar(g),\Delta_\hbar)$ a quantization of $(g,\delta_\hbar)$ through Etingof-Kazhdan functor. We prove the existence of a $L_\infty$-morphism between the Lie algebra $C(\g)=\Lambda(g)$ and…

Let g be a symmetrizable Kac-Moody algebra and U_h(g) its quantized enveloping algebra. The quantum Weyl group operators of U_h(g) and the universal R-matrices of its Levi subalgebras endow U_h(g) with a natural quasi-Coxeter…

Let g be a complex, semisimple Lie algebra. Drinfeld showed that the quantum loop algebra U_h(Lg) of g degenerates to the Yangian Y_h(g). We strengthen this result by constructing an explicit algebra homomorphism Phi defined over Q[[h]]…

We prove a tamely ramified version of the Kazhdan-Lusztig equivalence using factorization algebras. More precisely, we establish an equivalence between the DG category of Iwahori-integrable affine Lie algebra representations and the DG…