Bart Vlaar

University of Amsterdam, FNWI, Post-Doc

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Noel B. Salazar

KU Leuven

Andrej Dujella

University of Zagreb

David Seamon

Kansas State University

Prof. Dr. Raffaele Pisano, MSc, HDR (Habil.)

Université des Sciences et Technologies de Lille (Lille-1)

Ezinvi Baloïtcha

University of Abomey Calavi

Nicolás Andruskiewitsch

Universidad Nacional de Córdoba

Wagner Cortes

Universidade Federal do Rio Grande do Sul

Jan L Cieśliński

University of Bialystok

Nikos Mantzakouras

National & Kapodistrian University of Athens

Adnan Awad

The University Of Jordan

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Papers by Bart Vlaar

Tensor K-matrices for quantum symmetric pairs

arXiv (Cornell University), Feb 26, 2024

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Pseudo-symmetric pairs for Kac-Moody algebras

Hypergeometry, Integrability and Lie Theory

Lie algebra involutions and their fixed-point subalgebras give rise to symmetric spaces and real ... more Lie algebra involutions and their fixed-point subalgebras give rise to symmetric spaces and real forms of complex Lie algebras, and are well-studied in the context of symmetrizable Kac-Moody algebras. In this paper we study a generalization. Namely, we introduce the concept of a pseudo-involution, an automorphism which is only required to act involutively on a stable Cartan subalgebra, and the concept of a pseudo-fixed-point subalgebra, a natural substitute for the fixed-point subalgebra. In the symmetrizable Kac-Moody setting, we give a comprehensive discussion of pseudo-involutions of the second kind, the associated pseudo-fixed-point subalgebras, restricted root systems and Weyl groups, in terms of generalizations of Satake diagrams.

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Universal K-matrices for quantum Kac-Moody algebras

Representation Theory of The American Mathematical Society, Jul 19, 2022

We introduce the notion of a cylindrical bialgebra, which is a quasitriangular bialgebra H endowe... more We introduce the notion of a cylindrical bialgebra, which is a quasitriangular bialgebra H endowed with a universal K-matrix, i.e., a universal solution of a generalized reflection equation, yielding an action of cylindrical braid groups on tensor products of its representations. We prove that new examples of such universal K-matrices arise from quantum symmetric pairs of Kac-Moody type and depend upon the choice of a pair of generalized Satake diagrams. In finite type, this yields a refinement of a result obtained by Balagović and Kolb, producing a family of non-equivalent solutions interpolating between the quasi-K-matrix originally due to Bao and Wang and the full universal K-matrix. Finally, we prove that this construction yields formal solutions of the generalized reflection equation with a spectral parameter in the case of finite-dimensional representations over the quantum affine algebra U q Lsl 2 .

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Boundary Quantum Knizhnik–Zamolodchikov Equations and Bethe Vectors

Communications in Mathematical Physics, Nov 26, 2014

Solutions to boundary quantum Knizhnik-Zamolodchikov equations are constructed as bilateral sums ... more Solutions to boundary quantum Knizhnik-Zamolodchikov equations are constructed as bilateral sums involving "off-shell" Bethe vectors in case the reflection matrix is diagonal and only the 2-dimensional representation of Uq(sl 2) is involved. We also consider their rational and classical degenerations.

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$Research paper thumbnail of Solutions of the $U_q(\widehat{\mathfrak{sl}}_N)$ reflection equations$

Solutions of the $U_q(\widehat{\mathfrak{sl}}_N)$ reflection equations

White Rose Research Online (University of Leeds, The University of Sheffield, University of York), Mar 17, 2018

We find the complete set of invertible solutions of the untwisted and twisted reflection equation... more We find the complete set of invertible solutions of the untwisted and twisted reflection equations for the Bazhanov-Jimbo R-matrix of type A (1) N −1. We also show that all invertible solutions can be obtained by an appropriate affinization procedure from solutions of the constant untwisted and twisted reflection equations.

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A Q-operator for open spin chains II: boundary factorization

arXiv (Cornell University), Jan 10, 2023

In the algebraic approach to Baxter's Q-operators for the closed Heisenberg XXZ spin chain, certa... more In the algebraic approach to Baxter's Q-operators for the closed Heisenberg XXZ spin chain, certain infinite-dimensional 'prefundamental' representations of the q-deformed Borel subalgebra play a central role. To extend this formalism to open spin chains, one needs a factorization identity for particular solutions of the reflection equation associated to these representations. In the case of quantum affine sl2, we derive such an identity using the recent theory of universal K-matrices for quantum affine algebras.

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$Research paper thumbnail of A non-symmetric Yang-Baxter Algebra for the Quantum Nonlinear Schr\$

A non-symmetric Yang-Baxter Algebra for the Quantum Nonlinear Schr\

arXiv: Mathematical Physics, Jul 3, 2012

This thesis is submitted in accordance with the regulations for the degree of Doctor of Philosoph... more

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Quasi-bialgebras from set-theoretic type solutions of the Yang–Baxter equation

Letters in Mathematical Physics

We examine classes of quantum algebras emerging from involutive, non-degenerate set-theoretic sol... more We examine classes of quantum algebras emerging from involutive, non-degenerate set-theoretic solutions of the Yang–Baxter equation and their q-analogues. After providing some universal results on quasi-bialgebras and admissible Drinfeld twists, we show that the quantum algebras produced from set-theoretic solutions and their q-analogues are in fact quasi-triangular quasi-bialgebras. Specific illustrative examples compatible with our generic findings are worked out. In the q-deformed case of set-theoretic solutions, we also construct admissible Drinfeld twists similar to the set-theoretic ones, subject to certain extra constraints dictated by the q-deformation. These findings greatly generalize recent relevant results on set-theoretic solutions and their q-deformed analogues.

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$Research paper thumbnail of Quasitriangular coideal subalgebras of $U_q(\mathfrak{g})$ in terms of generalized Satake diagrams$

Quasitriangular coideal subalgebras of $U_q(\mathfrak{g})$ in terms of generalized Satake diagrams

Let $\mathfrak{g}$ be a finite-dimensional semisimple complex Lie algebra and $\theta$ an involut... more Let $\mathfrak{g}$ be a finite-dimensional semisimple complex Lie algebra and $\theta$ an involutive automorphism of $\mathfrak{g}$. It is well-known from works of Letzter, Kolb and Balagovi\'c that the fixed-point subalgebra $\mathfrak{k} = \mathfrak{g}^\theta$ has a quantum counterpart $B$, a coideal subalgebra of the Drinfeld-Jimbo quantum group $U_q(\mathfrak{g})$ possessing a cylinder-twisted universal K-matrix $\mathcal{K}$. The objects $\theta$, $\mathfrak{k}$, $B$ and $\mathcal{K}$ can all be described in terms of a combinatorial datum, a Satake diagram. In the present work we extend this construction to generalized Satake diagrams, objects first considered by Heck. A generalized Satake diagram defines a semisimple automorphism of $\mathfrak{g}$ restricting to the standard Cartan subalgebra $\mathfrak{h}$ as an involution. We show that it naturally leads to a subalgebra $\mathfrak{k}\subset \mathfrak{g}$, not necessarily a fixed-point subalgebra, but still satisfying $\m...

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

Let g be a complex simple Lie algebra. We prove that every finitedimensional representation of th... more Let g be a complex simple Lie algebra. We prove that every finitedimensional representation of the (untwisted) quantum affine algebra U q Lg 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 k Ă U q Lg. Moreover, we prove that every irreducible representation over U q Lg remains generically irreducible under restriction to U q 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. Contents 1. Introduction 1 2. Quantum affine algebras 6 3. Quantum affine symmetric pairs 12 4. Spectral K-matrices 18 5. Rational K-matrices 23 6. Generic restricted irreducibility 29 7. Solutions of the reflection equations 33 References 35

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Inhomogeneous boundary transfer matrices and boundary quantum KZ equations

Pseudo-symmetric pairs for Kac-Moody algebras

Involutive automorphisms of the second kind of Kac-Moody algebras and their fixed-point subalgebr... more Involutive automorphisms of the second kind of Kac-Moody algebras and their fixed-point subalgebras can be q-deformed following Letzter and Kolb. These q-deformed algebras play a major role in the theory of the reflection equation. Essentially the same constructions can be made in a larger setting, where the automorphism is required to act involutively only on a stable Cartan subalgebra. We give a comprehensive Lie- and Coxeter-theoretic discussion of pseudo-involutions, pseudo-fixed-point subalgebras, including a survey of the associated restricted root systems and Weyl groups in the Kac-Moody setting, in terms of generalizations of Satake diagrams.

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Boundary quantum Knizhnik-Zamolodchikov equations and Bethe vectors

Solutions to boundary quantum Knizhnik-Zamolodchikov equations are constructed as bilateral sums ... more Solutions to boundary quantum Knizhnik-Zamolodchikov equations are constructed as bilateral sums involving "off-shell" Bethe vectors in case the reflection matrix is diagonal and only the 2-dimensional representation of U_q(ŝl̂(̂2̂)̂) is involved. We also consider their rational and classical degenerations.

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Reflection matrices, coideal subalgebras and generalized Satake diagrams of affine type

We present a generalization of the theory of quantum symmetric pairs as developed by Kolb and Let... more We present a generalization of the theory of quantum symmetric pairs as developed by Kolb and Letzter. We introduce a class of generalized Satake diagrams that give rise to (not necessarily involutive) automorphisms of the second kind of symmetrizable Kac-Moody algebras g. These lead to right coideal subalgebras B_c,s of quantized enveloping algebras U_q(g). In the case that g is a twisted or untwisted affine Lie algebra of classical type Jimbo found intertwiners (equivariant maps) of the vector representation of U_q(g) yielding trigonometric solutions to the parameter-dependent quantum Yang-Baxter equation. In the present paper we compute intertwiners of the vector representation restricted to the subalgebras B_c,s when g is of type A^(1)_n, B^(1)_n, C^(1)_n and D^(1)_n. These intertwiners are matrix solutions to the parameter-dependent quantum reflection equation known as trigonometric reflection matrices. They are symmetric up to conjugation by a diagonal matrix and in many cases...

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Quasitriangular coideal subalgebras of U_q(g) in terms of generalized Satake diagrams

Let g be a finite-dimensional semisimple complex Lie algebra and θ an involutive automorphism of ... more Let g be a finite-dimensional semisimple complex Lie algebra and θ an involutive automorphism of g. According to Letzter, Kolb and Balagović the fixed-point subalgebra k = g^θ has a quantum counterpart B, a coideal subalgebra of the Drinfeld-Jimbo quantum group U_q(g) possessing a universal K-matrix K. The objects θ, k, B and K can all be described in terms of combinatorial datum, a Satake diagram. In the present work we extend this construction to generalized Satake diagrams, objects first considered by Heck. A generalized Satake diagram naturally defines a semisimple automorphism θ of g restricting to the standard Cartan subalgebra h as an involution. It also defines a subalgebra k⊂g satisfying k∩h = h^θ, but not necessarily a fixed-point subalgebra. The subalgebra k can be quantized to a coideal subalgebra of U_q(g) endowed with a universal K-matrix in the sense of Kolb and Balagović. We conjecture that all such coideal subalgebras of U_q(g) arise from generalized Satake diagrams...

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Boundary transfer matrices and boundary quantum KZ equations

A simple relation between inhomogeneous transfer matrices and boundary quantum KZ equations is ex... more A simple relation between inhomogeneous transfer matrices and boundary quantum KZ equations is exhibited for quantum integrable systems with reflecting boundary conditions, analogous to an observation by Gaudin for periodic systems. Thus the boundary quantum KZ equations receive a new motivation. We also derive the commutativity of Sklyanin's boundary transfer matrices by merely imposing appropriate reflection equations, i.e. without using the conditions of crossing symmetry and unitarity of the R-matrix.

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Solutions of the U_q(sl_N) reflection equations

We find the complete set of invertible solutions of the untwisted and twisted reflection equation... more We find the complete set of invertible solutions of the untwisted and twisted reflection equations for the Bazhanov-Jimbo R-matrix of type ${\mathrm A}^{(1)}_{N-1}$. We also show that all invertible solutions can be obtained by an appropriate affinization procedure from solutions of the constant untwisted and twisted reflection equations.

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Integral solutions to boundary quantum Knizhnik-Zamolodchikov equations

We construct integral representations of solutions to the boundary quantum Knizhnik-Zamolodchikov... more We construct integral representations of solutions to the boundary quantum Knizhnik-Zamolodchikov equations. These are difference equations taking values in tensor products of Verma modules of quantum affine sl_2, with the K-operators acting diagonally. The integrands in question are products of scalar-valued elliptic weight functions with vector-valued trigonometric weight functions (boundary Bethe vectors). These integrals give rise to a basis of solutions of the boundary qKZ equations over the field of quasi-constant meromorphic functions in weight subspaces of the tensor product.

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Boundary quantum Knizhnik-Zamolodchikov equations and fusion

In this paper we extend our previous results concerning Jackson integral solutions of the boundar... more In this paper we extend our previous results concerning Jackson integral solutions of the boundary quantum Knizhnik-Zamolodchikov equations with diagonal K-operators to higher-spin representations of quantum affine sl_2. First we give a systematic exposition of known results on R-operators acting in the tensor product of evaluation representations in Verma modules over quantum sl_2. We develop the corresponding fusion of K-operators, which we use to construct diagonal K-operators in these representations. We construct Jackson integral solutions of the associated boundary quantum Knizhnik-Zamolodchikov equations and explain how in the finite-dimensional case they can be obtained from our previous results by the fusion procedure.

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A non-symmetric Yang-Baxter Algebra for the Quantum Nonlinear Schrödinger Model

We study certain non-symmetric wavefunctions associated to the quantum nonlinear Schr\"oding... more We study certain non-symmetric wavefunctions associated to the quantum nonlinear Schr\"odinger model, introduced by Komori and Hikami using Gutkin's propagation operator, which involves representations of the degenerate affine Hecke algebra. We highlight how these functions can be generated using a vertex-type operator formalism similar to the recursion defining the symmetric (Bethe) wavefunction in the quantum inverse scattering method. Furthermore, some of the commutation relations encoded in the Yang-Baxter equation for the relevant monodromy matrix are generalized to the non-symmetric case.

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Tensor K-matrices for quantum symmetric pairs

arXiv (Cornell University), Feb 26, 2024

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Pseudo-symmetric pairs for Kac-Moody algebras

Hypergeometry, Integrability and Lie Theory

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Universal K-matrices for quantum Kac-Moody algebras

Representation Theory of The American Mathematical Society, Jul 19, 2022

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Boundary Quantum Knizhnik–Zamolodchikov Equations and Bethe Vectors

Communications in Mathematical Physics, Nov 26, 2014

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$Research paper thumbnail of Solutions of the $U_q(\widehat{\mathfrak{sl}}_N)$ reflection equations$

Solutions of the $U_q(\widehat{\mathfrak{sl}}_N)$ reflection equations

White Rose Research Online (University of Leeds, The University of Sheffield, University of York), Mar 17, 2018

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A Q-operator for open spin chains II: boundary factorization

arXiv (Cornell University), Jan 10, 2023

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$Research paper thumbnail of A non-symmetric Yang-Baxter Algebra for the Quantum Nonlinear Schr\$

A non-symmetric Yang-Baxter Algebra for the Quantum Nonlinear Schr\

arXiv: Mathematical Physics, Jul 3, 2012

This thesis is submitted in accordance with the regulations for the degree of Doctor of Philosoph... more

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Quasi-bialgebras from set-theoretic type solutions of the Yang–Baxter equation

Letters in Mathematical Physics

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$Research paper thumbnail of Quasitriangular coideal subalgebras of $U_q(\mathfrak{g})$ in terms of generalized Satake diagrams$

Quasitriangular coideal subalgebras of $U_q(\mathfrak{g})$ in terms of generalized Satake diagrams

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

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Inhomogeneous boundary transfer matrices and boundary quantum KZ equations

Pseudo-symmetric pairs for Kac-Moody algebras

Download

Boundary quantum Knizhnik-Zamolodchikov equations and Bethe vectors

Solutions to boundary quantum Knizhnik-Zamolodchikov equations are constructed as bilateral sums ... more Solutions to boundary quantum Knizhnik-Zamolodchikov equations are constructed as bilateral sums involving "off-shell" Bethe vectors in case the reflection matrix is diagonal and only the 2-dimensional representation of U_q(ŝl̂(̂2̂)̂) is involved. We also consider their rational and classical degenerations.

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Reflection matrices, coideal subalgebras and generalized Satake diagrams of affine type

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Quasitriangular coideal subalgebras of U_q(g) in terms of generalized Satake diagrams

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Boundary transfer matrices and boundary quantum KZ equations

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Solutions of the U_q(sl_N) reflection equations

We find the complete set of invertible solutions of the untwisted and twisted reflection equation... more We find the complete set of invertible solutions of the untwisted and twisted reflection equations for the Bazhanov-Jimbo R-matrix of type ${\mathrm A}^{(1)}_{N-1}$. We also show that all invertible solutions can be obtained by an appropriate affinization procedure from solutions of the constant untwisted and twisted reflection equations.

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Integral solutions to boundary quantum Knizhnik-Zamolodchikov equations

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Boundary quantum Knizhnik-Zamolodchikov equations and fusion

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A non-symmetric Yang-Baxter Algebra for the Quantum Nonlinear Schrödinger Model

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