Quantization of continuum Kac-Moody algebras

Continuum Kac-Moody algebras have been recently introduced by the authors and O. Schiffmann. These are Lie algebras governed by a continuum root system, which can be realized as uncountable colimits of Borcherds-Kac-Moody algebras. In this paper, we prove that any continuum Kac-Moody algebra is canonically endowed with a non-degenerate invariant bilinear form. The positive and negative Borel subalgebras form a Manin triple with respect to this pairing, inducing on the continuum Kac-Moody algebra a topological quasi-triangular Lie bialgebra structure. We then construct an explicit quantization, which we refer to as a continuum quantum group, and we show that the latter is similarly realized as an uncountable colimit of Drinfeld-Jimbo quantum groups.Comment: Final version. Minor change

Monodromy of the Casimir connection of a symmetrisable Kac-Moody algebra

Let g be a symmetrisable Kac-Moody algebra and V an integrable g-module in category O. We show that the monodromy of the (normally ordered) rational Casimir connection on V can be made equivariant with respect to the Weyl group W of g, and therefore defines an action of the braid group B_W of W on V. We then prove that this action is uniquely equivalent to the quantum Weyl group action of B_W on a quantum deformation of V, that is an integrable, category O-module V_h over the quantum group U_h(g) such that V_h/hV_h is isomorphic to V. This extends a result of the second author which is valid for g semisimple.Comment: One reference added. 48 page

Quasi-Coxeter categories and a relative Etingof-Kazhdan quantization functor

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 quasitriangular quasibialgebra structure which underlies the action of the braid group of g and Artin's braid groups on the tensor product of integrable, category O modules. We show that this structure can be transferred to the universal enveloping algebra Ug[[h]]. The proof relies on a modification of the Etingof-Kazhdan quantization functor, and yields an isomorphism between (appropriate completions of) U_h(g) and Ug[[h]] preserving a given chain of Levi subalgebras. We carry it out in the more general context of chains of Manin triples, and obtain in particular a relative version of the Etingof-Kazhdan functor with input a split pair of Lie bialgebras. Along the way, we develop the notion of quasi-Coxeter categories, which are to generalized braid groups what braided tensor categories are to Artin's braid groups. This leads to their succint description as a 2-functor from a 2-category whose morphisms are De Concini-Procesi associahedra. These results will be used in the sequel to this paper to give a monodromic description of the quantum Weyl group operators of an affine Kac-Moody algebra, extending the one obtained by the second author for a semisimple Lie algebra.Comment: 63 pages. Exposition in sections 1 and 4 improved. Material added: definition of a split pair of Lie bialgebras (sect. 5.2-5.5), 1-jet of the relative twist (5.20), PROP description of the Verma modules L_-,N*_+ (7.6), restriction to Levi subalgebras (8.4), D-structures on Kac-Moody algebras (9.1

Rational K-matrices for finite-dimensional representations of quantum affine algebras

Let $\mathfrak{g}$ be a complex simple Lie algebra. We prove that every finite-dimensional representation of the (untwisted) quantum affine algebra $U_qL\mathfrak{g}$ gives rise to a family of spectral K-matrices, namely solutions of Cherednik's generalized reflection equation, which depends upon the choice of a quantum affine symmetric pair $U_q\mathfrak{k}\subset U_qL\mathfrak{g}$. Moreover, we prove that every irreducible representation over $U_qL\mathfrak{g}$ remains generically irreducible under restriction to $U_q\mathfrak{k}$. From the latter result, we deduce that every obtained K-matrix can be normalized to a matrix-valued rational function in a multiplicative parameter, known in the study of quantum integrability as a trigonometric K-matrix. Finally, we show that our construction recovers many of the known solutions of the standard reflection equation and gives rise to a large class of new solutions.Comment: 37 page

Universal k-matrices for quantum Kac-Moody algebras

We define the notion of an \emph{almost cylindrical} bialgebra, which is roughly a quasitriangular bialgebra endowed with a universal solution of a {twisted} reflection equation, called a {twisted} universal k--matrix, yielding an action of cylindrical braid groups on tensor products of its representations. The definition is a nontrivial generalization of the notion of cylinder--braided bialgebras due to tom Dieck--H\"{a}ring-Oldenburg and Balagovi\'{c}--Kolb. Namely, the twisting involved in the reflection equation does not preserve the quasitriangular structure. Instead, it is only required to be an algebra automorphism, whose defect in being a morphism of quasitriangular bialgebras is controlled by a Drinfeld twist. We prove that examples of such new twisted universal k--matrices arise from quantum symmetric pairs of Kac--Moody type, whose controlling combinatorial datum is a pair of compatible generalized Satake diagrams. In finite type, this yields a refinement of the result obtained by Balagovi\'c--Kolb, producing a family of inequivalent solutions interpolating between the \emph{quasi}--k--matrix and the {\em full} universal k--matrix. This new framework is motivated by the study of solutions of the parameter--dependent reflection equation (spectral k--matrices) in the category of finite--dimensional representations of quantum affine algebras. Indeed, as an application, we prove that our construction leads to (formal) spectral k--matrices in evaluation representations of $U_qL{\mathfrak{sl}_2}$.Comment: 67 pages; made minor changes throughout the documen

Generalized Schur-Weyl dualities for quantum affine symmetric pairs and orientifold KLR algebras

We define a boundary analogue of the Kang-Kashiwara-Kim-Oh generalized Schur-Weyl dualities between quantum affine algebras and Khovanov-Lauda-Rouquier (KLR) algebras. Let $\mathfrak{g}$ be a complex simple Lie algebra and $U_qL\mathfrak{g}$ the corresponding quantum affine algebra. We construct a functor ${}^{\theta}{\sf F}$ between finite-dimensional modules over a quantum symmetric pair of affine type $U_q\mathfrak{k}\subset U_qL{\mathfrak{g}}$ and an orientifold KLR ($o$KLR) algebra arising from a framed quiver with a contravariant involution, whose nodes are indexed by finite-dimensional $U_qL{\mathfrak{g}}$-modules. With respect to the Kang-Kashiwara-Kim-Oh construction, our combinatorial model is further enriched with the poles of the trigonometric K-matrices (that is trigonometric solutions of a generalized reflection equation) intertwining the action of $U_q\mathfrak{k}$ on finite-dimensional $U_qL{\mathfrak{g}}$-modules. By construction, ${}^{\theta}{\sf F}$ is naturally compatible with the Kang-Kashiwara-Kim-Oh functor in that, while the latter is a functor of monoidal categories, ${}^{\theta}{\sf F}$ is a functor of module categories. Relying on an isomorphism between suitable completions of $o$KLR algebras and affine Hecke algebras of type $\sf C$, we prove that ${}^{\theta}{\sf F}$ recovers the Schur-Weyl dualities due to Fan-Lai-Li-Luo-Wang-Watanabe in quasi-split type $\sf AIII$. Finally, we construct spectral K-matrices for orientifold KLR algebras, yielding a meromorphic braiding on its category of finite-dimensional representations. We prove that, in the case of the ${\sf A}_{\infty}$ quiver with no fixed points and no framing, the functor ${}^{\theta}{\sf F}$ is exact, factors through a suitable localization, and takes values in a boundary analogue of the Hernandez-Leclerc category.Comment: 61 page

An explicit isomorphism between quantum and classical sl(n)

Let g be a complex, semisimple Lie algebra. Drinfeld showed that the quantum group associated to g is isomorphic as an algebra to the trivial deformation of the universal enveloping algebra of g. In this paper we construct explicitly such an isomorphism when g = sl(n), previously known only for n=2.Comment: 32 pages. Slightly modified introduction. To appear in Transformation Group

Uniqueness of Coxeter categories for Kac-Moody algebras

Let g be a symmetrisable Kac-Moody algebra, and U_h(g) the corresponding quantum group. We showed in arXiv:1610.09744 and arXiv:1610.09741 that the braided quasi-Coxeter structure on integrable, category O representations of U_h(g) which underlies the R-matrix actions arising from the Levi subalgebras of U_h(g) and the quantum Weyl group action of the generalised braid group B_g can be transferred to integrable, category O representations of g. We prove in this paper that, up to unique equivalence, there is a unique such structure on the latter category with prescribed restriction functors, R--matrices, and local monodromies. This extends, simplifies and strengthens a similar result of the second author valid when g is semisimple, and is used in arXiv:1512.03041 to describe the monodromy of the rational Casimir connection of g in terms of the quantum Weyl group operators of U_h(g). Our main tool is a refinement of Enriquez's universal algebras, which is adapted to the PROP describing a Lie bialgebra graded by the non-negative roots of g.Comment: Expanded Introduction and Sec. 5 to discuss convolution product (5.11), cosimplicial structure on basis elements (5.13) and module structure on coinvariants (5.15). Minor revisions in Sec. 7.1 (gradings), 7.4 (deformation DY modules), 9.7 (exposition), 15.7 (Drinfeld double) and 15.15 (rigidity for diagrammatic KM algebras). Final version, to appear in Adv. Math. 81 page