Koen Schouten | Searcharxiv

arXiv2026

The quantum group structure of long-range integrable deformations

Quantum integrable spin chains are known to possess a large family of long-range deformations generated by the local, boost and bilocal operators. Although these deformations are well-understood on the level of the pairwise commuting charges, the underlying quantum group structures had not yet been recognised. In this paper, we provide a quantum group-theoretical description for the family of long-range deformations of arbitrary homogeneous Yang-Baxter integrable spin chains up to first order in the deformation parameter. In particular, we show that the deformations are obtained via a twist of the algebraic structure of the underlying quantum group. This twisting results in a generally non-associative algebra that has a non-trivial Drinfeld associator. The Drinfeld associator is then shown to encode the information about the long-range interaction terms for the integrable spin chain. Moreover, the deformed quantum group is shown to contain a large perturbatively associative substructure, thus ensuring the perturbative integrability of the long-range model. The deformed quantum group provides explicit expressions for the Lax operators and R-matrices of the long-range deformed models, which manifestly satisfy the RLL relation and the Yang-Baxter equation up to first order in the deformation parameter. In order to derive the results, we introduce algebra elements that we call the algebraic charge densities. As a side result, we provide a conjecture for the explicit expressions of the undeformed charge densities in terms of these algebra elements.

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arXiv2025

The von Neumann algebraic quantum group $\mathrm{SU}_q(1,1)\rtimes \mathbb{Z}_2$ and the DSSYK model

The double-scaling limit of the SYK (DSSYK) model is known to possess an underlying $\mathcal{U}_q(\mathfrak{su}(1,1))$ quantum group symmetry. In this paper, we provide, for the first time, a von Neumann algebraic quantum group-theoretical description of the degrees of freedom and the dynamics of the DSSYK model. In particular, we construct the operator-algebraic quantum Gauss decomposition for the von Neumann algebraic quantum group $\mathrm{SU}_q(1,1)\rtimes \mathbb{Z}_2$, i.e. the $q$-deformation of the normaliser of $\mathrm{SU}(1,1)$ in $\mathrm{SL}(2,\mathbb{C})$, and derive the Casimir action on its quantum homogeneous spaces. We then show that the dynamics on quantum AdS$_{2,q}$ space reduces to that of the DSSYK model. Furthermore, we argue that the extension of the global symmetry group to its normaliser is not only necessary for a consistent definition of the locally compact quantum group, but that, moreover, the reduction to the DSSYK model works exclusively at the level of the normaliser. The von Neumann algebraic description is shown to give a natural restriction on the allowed quantised coordinates, elegantly ensuring length positivity and non-negative integer chord numbers. Lastly, we make remarks on the correlation function related to the strange series representation, which is argued to interpolate between the AdS and dS regions of our $q$-homogeneous space.

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