377 research outputs found
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
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
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 be a complex simple
Lie algebra and the corresponding quantum affine algebra. We
construct a functor between finite-dimensional modules
over a quantum symmetric pair of affine type and an orientifold KLR (KLR) algebra arising from a
framed quiver with a contravariant involution, whose nodes are indexed by
finite-dimensional -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
on finite-dimensional -modules. By
construction, is naturally compatible with the
Kang-Kashiwara-Kim-Oh functor in that, while the latter is a functor of
monoidal categories, is a functor of module categories.
Relying on an isomorphism between suitable completions of KLR algebras and
affine Hecke algebras of type , we prove that
recovers the Schur-Weyl dualities due to Fan-Lai-Li-Luo-Wang-Watanabe in
quasi-split type . 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 quiver with no fixed points and no framing, the functor
is exact, factors through a suitable localization, and
takes values in a boundary analogue of the Hernandez-Leclerc category.Comment: 61 page
Rational K-matrices for finite-dimensional representations of quantum affine algebras
Let be a complex simple Lie algebra. We prove that every
finite-dimensional representation of the (untwisted) quantum affine algebra
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 . Moreover, we prove that every irreducible representation
over remains generically irreducible under restriction to
. 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
.Comment: 67 pages; made minor changes throughout the documen
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
Coxeter categories and quantum groups
We define the notion of braided Coxeter category, which is informally a
tensor category carrying compatible, commuting actions of a generalised braid
group B_W and Artin's braid groups B_n on the tensor powers of its objects. The
data which defines the action of B_W bears a formal similarity to the
associativity constraints in a monoidal category, but is related to the
coherence of a family of fiber functors. We show that the quantum Weyl group
operators of a quantised Kac-Moody algebra U_h(g), together with the universal
R-matrices of its Levi subalgebras, give rise to a braided Coxeter structure on
integrable, category O-modules for U_h(g). By relying on the 2-categorical
extension of Etingof-Kazhdan quantisation obtained in arXiv:1610.09744, we then
prove that this structure can be transferred to integrable, category
O-representations of g. These results are used in arXiv:1512.03041 to give a
monodromic description of the quantum Weyl group operators of U_h(g) which
extends the one obtained by the second author for a semisimple Lie algebra.Comment: Substantial revision. Changes include 1) greatly expanded
introduction 2) Section 8 split into the new sections 8 (Universal Algebras)
and 9 (Universal pre-Coxeter Structures) 3) more thorough account of the
PROPic nature of transferred structure (Sect. 10) 4) notion of Upsilon and
a-strict pre-Coxeter structures (Sect. 3, 9 and 10). Final version. To appear
in Selecta Math. 81 page
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