An Askey-Wilson Algebra of Rank 2

Groenevelt, Wolter; Carel, Wagenaar,

doi:10.3842/sigma.2023.008

Cited by 4 publications

(2 citation statements)

References 33 publications

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“…To our knowledge there is only one example of models of this type. This is the dynamic-ASEP, recently introduced in the literature [4,23] for which duality results have been proven. This is a generalization of ASEP (asymmetric exclusion process) for which the interaction of a particle with the rest of the system depends on the number of particles at its right (or left), i.e.…”

Section: The Modelmentioning

confidence: 99%

Duality for a boundary driven asymmetric model of energy transport

Carinci,

Casini,

Franceschini

2024

J. Phys. A: Math. Theor.

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We study the Asymmetric Brownian Energy, a model of heat conduction deﬁned on the one-dimensional ﬁnite lattice with open boundaries. The system is shown to be dual to the Symmetric inclusion process with absorbing boundaries. The proof relies on a non-local map transformation procedure relating the model to its symmetric version. As an application, we show how the duality relation can be used to analytically compute suitable exponential moments with respect to the stationary measure.

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Section: The Modelmentioning

confidence: 99%

Duality for a boundary driven asymmetric model of energy transport

Carinci,

Casini,

Franceschini

2024

J. Phys. A: Math. Theor.

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“…Due to the importance of aw (3), different attempts to generalise its definition appeared previously for n = 4 [19,27] or for any n [10,11,12] but a complete set of relations had not been provided. The point of view in the paper is that it would be interesting to have an Askey-Wilson algebra aw(n), which would be a genuine generalisation of aw (3), and which would have the special algebra saw(n) as a quotient.…”

Section: Introductionmentioning

confidence: 99%

The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms

Crampé,

Frappat,

Poulain d'Andecy

et al. 2023

SIGMA

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We propose a definition by generators and relations of the rank $n-2$ Askey-Wilson algebra $\mathfrak{aw}(n)$ for any integer $n$, generalising the known presentation for the usual case $n=3$. The generators are indexed by connected subsets of $\{1,\dots,n\}$ and the simple and rather small set of defining relations is directly inspired from the known case of $n=3$. Our first main result is to prove the existence of automorphisms of $\mathfrak{aw}(n)$ satisfying the relations of the braid group on $n+1$ strands. We also show the existence of coproduct maps relating the algebras for different values of $n$. An immediate consequence of our approach is that the Askey-Wilson algebra defined here surjects onto the algebra generated by the intermediate Casimir elements in the $n$-fold tensor product of the quantum group ${\rm U}_q(\mathfrak{sl}_2)$ or, equivalently, onto the Kauffman bracket skein algebra of the $(n+1)$-punctured sphere. We also obtain a family of central elements of the Askey-Wilson algebras which are shown, as a direct by-product of our construction, to be sent to $0$ in the realisation in the $n$-fold tensor product of ${\rm U}_q(\mathfrak{sl}_2)$, thereby producing a large number of relations for the algebra generated by the intermediate Casimir elements.

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A generalized dynamic asymmetric exclusion process: orthogonal dualities and degenerations

Groenevelt,

Wagenaar

2024

J. Phys. A: Math. Theor.

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In this paper, a generalized version of dynamic ASEP is introduced, and it is shown that the process has a Markov duality property with the same process on the reversed lattice. The duality functions are multivariate q-Racah polynomials, and the corresponding orthogonality measure is the reversible measure of the process. By taking limits in the generator of dynamic ASEP, its reversible measure, and the duality functions, we obtain orthogonal and triangular dualities for several other interacting particle systems. In this sense, the duality of dynamic ASEP sits on top of a hierarchy of many dualities.

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An Askey-Wilson Algebra of Rank 2

Cited by 4 publications

References 33 publications

Duality for a boundary driven asymmetric model of energy transport

Duality for a boundary driven asymmetric model of energy transport

The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms

A generalized dynamic asymmetric exclusion process: orthogonal dualities and degenerations

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