Commutative subalgebras from Serre relations

Abstract

We demonstrate that commutativity of numerous one-dimensional subalgebras in $W_{1+\infty }$ algebra, i.e. the existence of many non-trivial integrable systems described in recent arXiv:2303.05273 follows from the subset of relations in algebra known as Serre relations. No other relations are needed for commutativity. The Serre relations survive the deformation to the affine Yangian $Y({\widehat{\mathfrak{gl}}}_1)$, hence the commutative subalgebras do as well. A special case of the Yangian parameters corresponds to the $\beta$-deformation. The preservation of Serre relations can be thought of a selection rule for proper systems of commuting $\beta$-deformed Hamiltonians. On the contrary, commutativity in the extended family associated with ``rational (non-integer) rays" is {\it not} reduced to the Serre relations, and uses also other relations in the $W_{1+\infty }$ algebra. Thus their $\beta$-deformation is less straightforward.

13 pages

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• Two $\beta$-ensemble realization of $\beta$-deformed WLZZ models
• Simple Representations of BPS Algebras: the case of $Y({\widehat{\mathfrak{gl}}}_2)$
• On the status of DELL systems
• Wall-Crossing Effects on Quiver BPS Algebras
• Algorithms for representations of quiver Yangian algebras
• Chalykh's Baker-Akhiezer functions as eigenfunctions of the integer-ray integrable systems
• $W_{1+\infty }$ and $\tilde{W}$ algebras, and Ward identities
• On bilinear superintegrability for monomial matrix models in pure phase
• Deformation of superintegrability in the Miwa-deformed Gaussian matrix model
• On character expansion and Gaussian regularization of Itzykson-Zuber measure
• Generalized $\tilde{W}$ algebras
• Tunnels Under Geometries (or Instantons Know Their Algebras)
• CMM formula as superintegrability property of unitary model
• Twisted Cherednik spectrum as a $q,t$-deformation
• Weyl Mutations in Quiver Yangians