Bart Vlaar
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Bart Vlaar has a total of 11 co-authors. Here are the top co-authors by number of shared publications:
- Vidas Regelskis
University of Hertfordshire
5 shared publications - Jasper V. Stokman
KdV Institute for Mathematics, University of Amsterdam, Amsterdam, The Netherlands
4 shared publications - Nicolai Reshetikhin
Beijing Institute for Mathematical Sciences and Applications, Beijing, China
4 shared publications - Andrea Appel
School of Mathematics, University of Edinburgh, Edinburgh, UK
3 shared publications - Robert Weston
Department of Mathematics Heriot-Watt University EH14 4AS Edinburgh UK
3 shared publications - Alec Cooper
Department of Mathematics, Heriot-Watt University, Edinburgh, Scotland, UK
2 shared publications
Latest Publications
- Top Cited Papers
- 11 August 2025
- Published by European Mathematical Society - EMS - Publishing House GmbH in Journal of the European Mathematical Society
Trigonometric K-matrices for finite-dimensional representations of quantum affine algebras
Let \mathfrak{g} be a complex simple finite-dimensional Lie algebra and U_{q}(\widehat{\mathfrak{g}}) the corresponding quantum affine algebra. We prove that every irreducible finite-dimensional U_{q}(\widehat{\mathfrak{g}}) -module gives rise to a family of trigonometric solutions of Cherednik’s generalized reflection equation. These depend on the choice of a quantum affine symmetric pair U_{q}(\mathfrak{k})\subset U_{q}(\widehat{\mathfrak{g}}) . Our result relies on the construction of universal K-matrices for arbitrary quantum symmetric pairs we obtained in [Represent. Theory 26 , 764–824 (2022)] and the fact that every irreducible U_{q}(\widehat{\mathfrak{g}}) -module is generically irreducible under restriction to U_{q}(\mathfrak{k}) . In the case of small modules and Kirillov–Reshetikhin modules, we obtain new solutions of the standard and the transposed reflection equations.
- 11 August 2025
- Published by European Mathematical Society - EMS - Publishing House GmbH in Journal of the European Mathematical Society
Top Cited PapersTrigonometric K-matrices for finite-dimensional representations of quantum affine algebras
Let \mathfrak{g} be a complex simple finite-dimensional Lie algebra and U_{q}(\widehat{\mathfrak{g}}) the corresponding quantum affine algebra. We prove that every irreducible finite-dimensional U_{q}(\widehat{\mathfrak{g}}) -module gives rise to a family of trigonometric solutions of Cherednik’s generalized reflection equation. These depend on the choice of a quantum affine symmetric pair U_{q}(\mathfrak{k})\subset U_{q}(\widehat{\mathfrak{g}}) . Our result relies on the construction of universal K-matrices for arbitrary quantum symmetric pairs we obtained in [Represent. Theory 26 , 764–824 (2022)] and the fact that every irreducible U_{q}(\widehat{\mathfrak{g}}) -module is generically irreducible under restriction to U_{q}(\mathfrak{k}) . In the case of small modules and Kirillov–Reshetikhin modules, we obtain new solutions of the standard and the transposed reflection equations.
- 6 June 2025
- Published by Elsevier in Indagationes Mathematicae
- Vol. 36 (6) , 1830-1862
Boundary transfer matrices arising from quantum symmetric pairs
- 6 June 2025
- Published by Elsevier in Indagationes Mathematicae
- Vol. 36 (6) , 1830-1862
Boundary transfer matrices arising from quantum symmetric pairs
- Open Access
- 4 April 2025
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 406 (5) , 1-57
Tensor K-Matrices for Quantum Symmetric Pairs
Let $${{\mathfrak {g}}}$$ be a symmetrizable Kac–Moody algebra, $$U_q({{\mathfrak {g}}})$$ its quantum group, and $$U_q({\mathfrak {k}})\subset U_q({{\mathfrak {g}}})$$ a quantum symmetric pair subalgebra determined by a Lie algebra automorphism $$\theta $$ . We introduce a category $$\mathcal {W}_{\theta }$$ of weight $$U_q({\mathfrak {k}})$$ -modules, which is acted on by the category of weight $$U_q({{\mathfrak {g}}})$$ -modules via tensor products. We construct a universal tensor K-matrix $${{\mathbb {K}}} $$ (that is, a solution of a reflection equation) in a completion of $$U_q({\mathfrak {k}})\otimes U_q({{\mathfrak {g}}})$$ . This yields a natural operator on any tensor product $$M\otimes V$$ , where $$M\in \mathcal {W}_{\theta }$$ and $$V\in {{\mathcal {O}}}_\theta $$ , i.e., V is a $$U_q({{\mathfrak {g}}})$$ -module in category $${{\mathcal {O}}}$$ satisfying an integrability property determined by $$\theta $$ . Canonically, $$\mathcal {W}_{\theta }$$ is equipped with a structure of a bimodule category over $${{\mathcal {O}}}_\theta $$ and the action of $${{\mathbb {K}}} $$ is encoded by a new categorical structure, which we call a boundary structure on $$\mathcal {W}_{\theta }$$ . This generalizes a result of Kolb which describes a braided module structure on finite-dimensional $$U_q({\mathfrak {k}})$$ -modules when $${{\mathfrak {g}}}$$ is finite-dimensional. We also consider our construction in the case of the category $${{\mathcal {C}}}$$ of finite-dimensional modules of a quantum affine algebra, providing the most comprehensive universal framework to date for large families of solutions of parameter-dependent reflection equations. In this case the tensor K-matrix gives rise to a formal Laurent series with a well-defined action on tensor products of any module in $$\mathcal {W}_{\theta }$$ and any module in $${{\mathcal {C}}}$$ . This series can be normalized to an operator-valued rational function, which we call trigonometric tensor K-matrix, if both factors in the tensor product are in $${{\mathcal {C}}}$$ .- 4 April 2025
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 406 (5) , 1-57
Open AccessTensor K-Matrices for Quantum Symmetric Pairs
Let $${{\mathfrak {g}}}$$ be a symmetrizable Kac–Moody algebra, $$U_q({{\mathfrak {g}}})$$ its quantum group, and $$U_q({\mathfrak {k}})\subset U_q({{\mathfrak {g}}})$$ a quantum symmetric pair subalgebra determined by a Lie algebra automorphism $$\theta $$ . We introduce a category $$\mathcal {W}_{\theta }$$ of weight $$U_q({\mathfrak {k}})$$ -modules, which is acted on by the category of weight $$U_q({{\mathfrak {g}}})$$ -modules via tensor products. We construct a universal tensor K-matrix $${{\mathbb {K}}} $$ (that is, a solution of a reflection equation) in a completion of $$U_q({\mathfrak {k}})\otimes U_q({{\mathfrak {g}}})$$ . This yields a natural operator on any tensor product $$M\otimes V$$ , where $$M\in \mathcal {W}_{\theta }$$ and $$V\in {{\mathcal {O}}}_\theta $$ , i.e., V is a $$U_q({{\mathfrak {g}}})$$ -module in category $${{\mathcal {O}}}$$ satisfying an integrability property determined by $$\theta $$ . Canonically, $$\mathcal {W}_{\theta }$$ is equipped with a structure of a bimodule category over $${{\mathcal {O}}}_\theta $$ and the action of $${{\mathbb {K}}} $$ is encoded by a new categorical structure, which we call a boundary structure on $$\mathcal {W}_{\theta }$$ . This generalizes a result of Kolb which describes a braided module structure on finite-dimensional $$U_q({\mathfrak {k}})$$ -modules when $${{\mathfrak {g}}}$$ is finite-dimensional. We also consider our construction in the case of the category $${{\mathcal {C}}}$$ of finite-dimensional modules of a quantum affine algebra, providing the most comprehensive universal framework to date for large families of solutions of parameter-dependent reflection equations. In this case the tensor K-matrix gives rise to a formal Laurent series with a well-defined action on tensor products of any module in $$\mathcal {W}_{\theta }$$ and any module in $${{\mathcal {C}}}$$ . This series can be normalized to an operator-valued rational function, which we call trigonometric tensor K-matrix, if both factors in the tensor product are in $${{\mathcal {C}}}$$ . - Open Access
- 25 April 2024
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 405 (5) , 1-42
A Q-Operator for Open Spin Chains II: Boundary Factorization
One of the features of Baxter’s Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to Q-operators, underlying this is a factorization formula for L-operators (solutions of the Yang–Baxter equation associated to particular infinite-dimensional representations). To extend such a formalism to open spin chains, one needs a factorization identity for solutions of the reflection equation (boundary Yang–Baxter equation) associated to these representations. In the case of quantum affine $$\mathfrak {sl}_2$$ and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras.- 25 April 2024
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 405 (5) , 1-42
Open AccessA Q-Operator for Open Spin Chains II: Boundary Factorization
One of the features of Baxter’s Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to Q-operators, underlying this is a factorization formula for L-operators (solutions of the Yang–Baxter equation associated to particular infinite-dimensional representations). To extend such a formalism to open spin chains, one needs a factorization identity for solutions of the reflection equation (boundary Yang–Baxter equation) associated to these representations. In the case of quantum affine $$\mathfrak {sl}_2$$ and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras. - Open Access
A Q-operator for open spin chains II: boundary factorization
One of the features of Baxter's Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to Q-operators, underlying this is a factorization formula for L-operators (solutions of the Yang-Baxter equation associated to particular infinite-dimensional representations). To have such a formalism to open spin chains, one needs a factorization identity for solutions of the reflection equation (boundary Yang-Baxter equation) associated to these representations. In the case of quantum affine $\mathfrak{sl}_2$ and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras.Open AccessA Q-operator for open spin chains II: boundary factorization
One of the features of Baxter's Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to Q-operators, underlying this is a factorization formula for L-operators (solutions of the Yang-Baxter equation associated to particular infinite-dimensional representations). To have such a formalism to open spin chains, one needs a factorization identity for solutions of the reflection equation (boundary Yang-Baxter equation) associated to these representations. In the case of quantum affine $\mathfrak{sl}_2$ and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras. - Open Access
Trigonometric K-matrices for finite-dimensional representations of quantum affine algebras
Let $\mathfrak{g}$ be a complex simple Lie algebra and $U_q\hat{\mathfrak{g}}$ the corresponding quantum affine algebra. We prove that every irreducible finite-dimensional $U_q\hat{\mathfrak{g}}$-module gives rise to a family of trigonometric K-matrices, i.e., rational solutions of Cherednik's generalized reflection equation. The result depends upon the choice of a quantum affine symmetric pair $U_q\mathfrak{k}\subset U_q\hat{\mathfrak{g}}$. It hinges on the construction of universal K-matrices for arbitrary quantum symmetric pairs, and relies on proving that every irreducible $U_q\hat{\mathfrak{g}}$-module is generically irreducible under restriction to $U_q{\mathfrak{k}}$. In the case of small modules and Kirillov-Reshetikhin modules, we obtain new solutions of the standard and the transposed reflection equations.Open AccessTrigonometric K-matrices for finite-dimensional representations of quantum affine algebras
Let $\mathfrak{g}$ be a complex simple Lie algebra and $U_q\hat{\mathfrak{g}}$ the corresponding quantum affine algebra. We prove that every irreducible finite-dimensional $U_q\hat{\mathfrak{g}}$-module gives rise to a family of trigonometric K-matrices, i.e., rational solutions of Cherednik's generalized reflection equation. The result depends upon the choice of a quantum affine symmetric pair $U_q\mathfrak{k}\subset U_q\hat{\mathfrak{g}}$. It hinges on the construction of universal K-matrices for arbitrary quantum symmetric pairs, and relies on proving that every irreducible $U_q\hat{\mathfrak{g}}$-module is generically irreducible under restriction to $U_q{\mathfrak{k}}$. In the case of small modules and Kirillov-Reshetikhin modules, we obtain new solutions of the standard and the transposed reflection equations. - Open Access
Quasi-bialgebras from set-theoretic type solutions of the Yang-Baxter equation
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 generalise recent relevant results on set theoretic solutions and their q-deformed analogues.
Open AccessQuasi-bialgebras from set-theoretic type solutions of the Yang-Baxter equation
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 generalise recent relevant results on set theoretic solutions and their q-deformed analogues.
- Open Access
Pseudo-symmetric pairs for Kac-Moody algebras
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.
Open AccessPseudo-symmetric pairs for Kac-Moody algebras
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.
- Open AccessQuasitriangular coideal subalgebras of $U_q(\mathfrak{g})$ in terms of generalized Satake diagramsLet $\mathfrak{g}$ be a finite-dimensional semisimple complex Lie algebra and $\theta$ an involutive automorphism of $\mathfrak{g}$. According to G. Letzter, S. Kolb and M. Balagovi\'c 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 universal K-matrix $\mathcal{K}$. The objects $\theta$, $\mathfrak{k}$, $B$ and $\mathcal{K}$ can all be described in terms of Satake diagrams. In the present work we extend this construction to generalized Satake diagrams, combinatorial data first considered by A. Heck. A generalized Satake diagram naturally defines a semisimple automorphism $\theta$ of $\mathfrak{g}$ restricting to the standard Cartan subalgebra $\mathfrak{h}$ as an involution. It also defines a subalgebra $\mathfrak{k}\subset \mathfrak{g}$ satisfying $\mathfrak{k} \cap \mathfrak{h} = \mathfrak{h}^\theta$, but not necessarily a fixed-point subalgebra. The subalgebra $\mathfrak{k}$ can be quantized to a coideal subalgebra of $U_q(\mathfrak{g})$ endowed with a universal K-matrix in the sense of Kolb and Balagovi\'c. We conjecture that all such coideal subalgebras of $U_q(\mathfrak{g})$ arise from generalized Satake diagrams in this way.Open AccessQuasitriangular coideal subalgebras of $U_q(\mathfrak{g})$ in terms of generalized Satake diagramsLet $\mathfrak{g}$ be a finite-dimensional semisimple complex Lie algebra and $\theta$ an involutive automorphism of $\mathfrak{g}$. According to G. Letzter, S. Kolb and M. Balagovi\'c 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 universal K-matrix $\mathcal{K}$. The objects $\theta$, $\mathfrak{k}$, $B$ and $\mathcal{K}$ can all be described in terms of Satake diagrams. In the present work we extend this construction to generalized Satake diagrams, combinatorial data first considered by A. Heck. A generalized Satake diagram naturally defines a semisimple automorphism $\theta$ of $\mathfrak{g}$ restricting to the standard Cartan subalgebra $\mathfrak{h}$ as an involution. It also defines a subalgebra $\mathfrak{k}\subset \mathfrak{g}$ satisfying $\mathfrak{k} \cap \mathfrak{h} = \mathfrak{h}^\theta$, but not necessarily a fixed-point subalgebra. The subalgebra $\mathfrak{k}$ can be quantized to a coideal subalgebra of $U_q(\mathfrak{g})$ endowed with a universal K-matrix in the sense of Kolb and Balagovi\'c. We conjecture that all such coideal subalgebras of $U_q(\mathfrak{g})$ arise from generalized Satake diagrams in this way.
- Open Access
- 23 June 2020
- Published by Wiley in Bulletin of the London Mathematical Society
- Vol. 52 (4) , 693-715
Quasitriangular coideal subalgebras of Uq(g) in terms of generalized Satake diagrams
- 23 June 2020
- Published by Wiley in Bulletin of the London Mathematical Society
- Vol. 52 (4) , 693-715
Open AccessQuasitriangular coideal subalgebras of Uq(g) in terms of generalized Satake diagrams