1,720,961 research outputs found
Monodromy of the Casimir connection of a symmetrisable Kac-Moody algebra
Full text linkLet 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 canonically 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: Some details added. Minor revisions throughout. Published version.
107 page
Yangians, quantum loop algebras and elliptic quantum groups
No full textThe Yangian Yg and quantum loop algebra Uq(Lg) of a complex
semisimple Lie algebra g share very many similarities, and were
long thought to have the same representations, though no precise
relation between them existed until recently.
I will explain how to construct a faithful functor from the finite-dimensional
representations of Yg to those of Uq(Lg). The functor is entirely
explicit, and governed by the monodromy of the abelian difference
equations determined by the commuting fields of the Yangian. It
yields a meromorphic, braided Kazhdan-Lusztig equivalence between
finite-dimensional representations of the Yg and of U_q(Lg).
A similar construction yields a faithful functor from representations
of U_q(Lg) to those of the elliptic quantum group E_{q,t}(g) corresponding
to g. This allows in particular a classification of irreducible finite-dimensional
representations of E_{q,tau}(g), which was previously unknown.
This is joint work with Sachin Gautam (Perimeter Institute).Non UBCUnreviewedAuthor affiliation: Northeastern UniversityFacult
Flat Connections and Quantum Groups
No full text20 pages. To appear in the Proceedings of the 2000 Twente Conference on Lie Groups, in a special issue of Acta Applicandae MathematicaeWe review the Kohno-Drinfeld theorem as well as a conjectural analogue relating quantum Weyl groups to the monodromy of a flat connection D on the Cartan subalgebra of a complex, semi-simple Lie algebra g with poles on the root hyperplanes and values in any g-module V. We sketch our proof of this conjecture when g=sl(n) and when g is arbitrary and V is a vector, spin or adjoint representation. We also establish a precise link between the connection D and Cherednik's generalisation of the KZ connection to finite reflection groups
Cohomological construction of relative twists
No full textAbstractLet g be a complex, semi-simple Lie algebra, h⊂g a Cartan subalgebra and D a subdiagram of the Dynkin diagram of g. Let gD⊂lD⊆g be the corresponding semi-simple and Levi subalgebras and consider two invariant solutions Φ∈(Ug⊗3〚ℏ〛)g and ΦD∈(UgD⊗3〚ℏ〛)gD of the pentagon equation for g and gD respectively. Motivated by the theory of quasi-Coxeter quasitriangular quasibialgebras [V. Toledano Laredo, Quasi-Coxeter algebras, Dynkin diagram cohomology and quantum Weyl groups, math.QA/0506529], we study in this paper the existence of a relative twist, that is an element F∈(Ug⊗2〚ℏ〛)lD such that the twist of Φ by F is ΦD. Adapting the method of Donin and Shnider [J. Donin, S. Shnider, Cohomological construction of quantized universal enveloping algebras, Trans. Amer. Math. Soc. 349 (1997) 1611–1632], who treated the case of an empty D, so that lD=h and ΦD=1⊗3, we give a cohomological construction of such an F under the assumption that ΦD is the image of Φ under the generalised Harish-Chandra homomorphism (Ug⊗3)lD→(UgD⊗3)gD. We also show that F is unique up to a gauge transformation if lD is of corank 1 or F satisfies FΘ=F21 where Θ∈Aut(g) is an involution acting as −1 on h
On the Finkelberg-Ginzburg Mirabolic Monodromy Conjecture
Full text linkWe compute the monodromy of the mirabolic Harish-Chandra D-module for all
values of the parameters (theta,c) in rank 1, and outside an explicit
codimension 2 set of values in ranks 2 and higher. This shows in particular
that the Finkelberg-Ginzburg conjecture, which is known to hold for generic
values of (theta,c), fails at special values even in rank 1. Our main tools are
Opdam's shift operators and intertwiners for the extended affine Weyl group,
which allow for the resolution of resonances outside the codimension two set.Comment: Substantial revision. 36 pages, 7 figure
Pure braid group actions on category O modules
Full text linkLet g be a symmetrisable Kac-Moody algebra and U_h(g) its quantised
enveloping algebra. Answering a question of P. Etingof, we prove that the
quantum Weyl group operators of U_h(g) give rise to a canonical action of the
pure braid group of g on any category O (not necessarily integrable)
U_h(g)-module V. By relying on our recent results in arXiv:1512.03041, we show
that this action describes the monodromy of the rational Casimir connection on
the g-module corresponding to V under the Etingof-Kazhdan equivalence of
category O for g and U_h(g). We also extend these results to yield equivalent
quantum Weyl group and monodromic representations of parabolic pure braid
groups on parabolic category O for U_h(g) and g.Comment: Final version, to appear in PAMQ. 36 page
The trigonometric Casimir connection of a simple Lie algebra
No full textAbstractLet g be a complex, semisimple Lie algebra, G the corresponding simply-connected Lie group and H⊂G a maximal torus. We construct a flat connection on H with logarithmic singularities on the root hypertori and values in the Yangian Y(g) of g. By analogy with the rational Casimir connection of g, we conjecture that the monodromy of this trigonometric connection is described by the quantum Weyl group operators of the quantum loop algebra Uℏ(Lg)
Coxeter categories and quantum groups
Full text linkWe define the notion of braided Coxeter category, which is informally a monoidal 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_{\hbar }{{\mathfrak {g}}}, together with the universal R-matrices of its Levi subalgebras, give rise to a braided Coxeter category structure on integrable, category {\mathcal {O}}-modules for U_{\hbar }{{\mathfrak {g}}}. By relying on the 2-categorical extension of Etingof–Kazhdan quantisation obtained in Appel and Toledano Laredo (Selecta Math NS 24:3529–3617, 2018), we then prove that this structure can be transferred to integrable, category {\mathcal {O}}-representations of {\mathfrak {g}}. These results are used in Appel and Toledano Laredo (arXiv:1512.03041, p 48, 2015) to give a monodromic description of the quantum Weyl group operators of U_{\hbar }{{\mathfrak {g}}}, which extends the one obtained by the second author for a semisimple Lie algebra
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