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 $Y(\hat{\mathfrak{gl}}_1)$, 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.