arXiv subjects

V. Toledano-Laredo

Publications and source records attributed to V. Toledano-Laredo.

10 recordsLinked to original sources

arXiv2025

An abelian formula for the quantum Weyl group action of the coroot lattice

Let g be a complex simple Lie algebra and Uq(Lg) its quantum loop algebra, where q is not a root of unity. We give an explicit formula for the quantum Weyl group action of the coroot lattice Q of g on finite-dimensional representations of Uq(Lg) in terms of its commuting generators. The answer is expressed in terms of the Chari-Pressley series, whose evaluation on highest weight vectors gives rise to Drinfeld polynomials. It hinges on a strong rationality result for that series, which is derived in the present paper. As an application, we identify the action of Q on the equivariant K-theory of Nakajima quiver varieties with that of explicitly given determinant line bundles.

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arXiv2022

Stokes phenomena, Poisson-Lie groups and quantum groups

Let g be a complex semisimple Lie algebra, G the simply-connected Poisson-Lie group corresponding to g, and G* its dual. G-valued Stokes phenomena were used by Boalch [Bo1,Bo2] to give a canonical, analytic linearisation of the Poisson structure on G*. Ug-valued Stokes phenomena were used by the first author to construct a twist killing the KZ associator, and therefore give a transcendental construction of the Drinfeld-Jimbo quantum group U_hg (arXiv:1601.04076). In the present paper, we show that the former construction can be obtained as semiclassical limit of the latter. Along the way, we also show that the R-matrix of U_hg is a Stokes matrix for the dynamical KZ equations.

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arXiv2015

Yangians, quantum loop algebras and abelian difference equations

Let g be a complex, semisimple Lie algebra, and Y_h(g) and U_q(Lg) the Yangian and quantum loop algebra of g. Assuming that h is not a rational number and that q=exp(i πh), we construct an equivalence between the finite-dimensional representations of U_q(Lg) and an explicit subcategory of those of Y_h(g) defined by choosing a branch of the logarithm. This equivalence is governed by the monodromy of the abelian additive difference equations defined by the commuting fields of Y_h(g). Our results are compatible with q-characters, and apply more generally to a symmetrisable Kac-Moody algebra g, in particular to affine Yangians and quantum toroidal algebras. In this generality, they yield an equivalence between the representations of Y_h(g) and U_q(Lg) whose restriction to g and U_q(g) respectively are integrable and in category O.

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arXiv2011

Stokes factors and multilogarithms

Let G be a complex, affine algebraic group and D a meromorphic connection on the trivial G-bundle over P^1, with a pole of order 2 at zero and a pole of order 1 at infinity. We show that the map S taking the residue of D at zero to the corresponding Stokes factors is given by an explicit, universal Lie series whose coefficients are multilogarithms. Using a non-commutative analogue of the compositional inversion of formal power series, we show that the same holds for the inverse of S, and that the corresponding Lie series coincides with the generating function for counting invariants in abelian categories constructed by D. Joyce.

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arXiv2009

The Dynkin diagram cohomology of finite Coxeter groups

Let D be a connected graph. The Dynkin complex CD(A) of a D-algebra A was introduced by the second author in [TL2] to control the deformations of quasi-Coxeter algebra structures on A. In the present paper, we study the cohomology of this complex when A is the group algebra of a Coxeter group W and D is the Dynkin diagram of W. We compute this cohomology when W is finite and prove in particular the rigidity of quasi-Coxeter algebra structures on kW. For an arbitrary W, we compute the top cohomology group and obtain a number of additional partial results when W is affine. Our computations are carried out by filtering CD(A) by the number of vertices of subgraphs of D. The corresponding graded complex turns out to be dual to the sum of the Coxeter complexes of all standard, irreducible parabolic subgroups of W.

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arXiv2009

Gaudin models with irregular singularities

We introduce a class of quantum integrable systems generalizing the Gaudin model. The corresponding algebras of quantum Hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine Kac-Moody algebra at the critical level, extending the construction of higher Gaudin Hamiltonians from hep-th/9402022 to the case of non-highest weight representations of affine algebras. We show that these algebras are isomorphic to algebras of functions on the spaces of opers on P^1 with regular as well as irregular singularities at finitely many points. We construct eigenvectors of these Hamiltonians, using Wakimoto modules of critical level, and show that their spectra on finite-dimensional representations are given by opers with trivial monodromy. We also comment on the connection between the generalized Gaudin models and the geometric Langlands correspondence with ramification.

math.QA↗

arXiv2008

Quasi-Coxeter algebras, Dynkin diagram cohomology and quantum Weyl groups

The author, and independently De Concini, conjectured that the monodromy of the Casimir connection of a simple Lie algebra g is described by the quantum Weyl group operators of the quantum group U_h(g). The aim of this paper, and of its sequel [TL4], is to prove this conjecture. The proof relies upon the use of quasi-Coxeter algebras, which are to generalised braid groups what Drinfeld's quasitriangular quasibialgebras are to the Artin braid groups B_n. Using an appropriate deformation cohomology, we reduce the conjecture to the existence of a quasi-Coxeter, quasitriangular quasibialgebra structure on the enveloping algebra Ug which interpolates between the quasi-Coxeter structure underlying the Casimir connection and the quasitriangular quasibialgebra underlying the KZ equations. The existence of this structure will be proved in [TL4].

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arXiv2006

Cohomological construction of relative twists

Let g be a complex, semi-simple Lie algebra, h a Cartan subalgebra of g and D a subdiagram of the Dynkin diagram of g. Let g_D and l_D be the corresponding semi-simple and Levi subalgebras and consider two invariant solutions Phi, Phi_D of the pentagon equation for g and g_D respectively. Motivated by the theory of quasi-Coxeter quasitriangular quasibialgebras \cite{TL3}, we study in this paper the existence of a relative twist, that is an element F invariant under l_D such that the twist of Phi by F is Phi_D. Adapting the method of Donin and Shnider, who treated the case of an empty D, so that l_D=h and Phi_D=1, we give a cohomological construction of such an F under the assumption that Phi_D is the image of Phi under the generalised Harish-Chandra homomorphism. We also show that F is unique up to a gauge transformation if l_D is of corank 1 or F satisfies F^Θ= F^{21} where Θis an involution of g acting as -1 on h.

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arXiv2004

Fusion of Positive Energy Representations of LSpin(2n)

Building upon the Jones-Wassermann program of studying Conformal Field Theory using operator algebraic tools, and the work of A. Wassermann on the loop group of LSU(n) (Invent. Math. 133 (1998), 467-538), we give a solution to the problem of fusion for the loop group of Spin(2n). Our approach relies on the use of A. Connes' tensor product of bimodules over a von Neumann algebra to define a multiplicative operation (Connes fusion) on the (integrable) positive energy representations of a given level. The notion of bimodules arises by restricting these representations to loops with support contained in an interval I of the circle or its complement. We study the corresponding Grothendieck ring and show that fusion with the vector representation is given by the Verlinde rules. The computation rests on 1) the solution of a 6-parameter family of Knizhnik-Zamolodchikhov equations and the determination of its monodromy, 2) the explicit construction of the primary fields of the theory, which allows to prove that they define operator-valued distributions and 3) the algebraic theory of superselection sectors developed by Doplicher-Haag-Roberts.

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arXiv2004

Casimir Operators and Monodromy Representations of Generalised Braid Groups

Let g be a complex, simple Lie algebra with Cartan subalgebra h and Weyl group W. We construct a one-parameter family of flat connections D on h with values in any finite-dimensional h-module V and simple poles on the root hyperplanes. The corresponding monodromy representation of the braid group B of type g is a deformation of the action of (a finite extension of) W on V. The residues of D are the Casimirs of the sl(2)-subalgebras of g corresponding to its roots. The irreducibility of a subspace U of V under these implies that, for generic values of the parameter, the braid group B acts irreducibly on U. Answering a question of Knutson and Procesi, we show that these Casimirs act irreducibly on the weight spaces of all simple g-modules if g is sl(3) but that this is not the case if g is not isomorphic to sl(2) or sl(3). We use this to disprove a conjecture of Kwon and Lusztig stating the irreducibility of quantum Weyl group actions of Artin's braid group B_n on the zero weight spaces of all simple U_{q}sl(n)-modules for n greater or equal to 4. Finally, we study the irreducibility of the action of the Casimirs on the zero weight spaces of self-dual g-modules and obtain complete classification results for g=sl(n) or g(2).

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