Commutative subalgebras from Serre relations

Abstract

We demonstrate that commutativity of numerous one-dimensional subalgebras in W1+∞ 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(glˆ1), hence the commutative subalgebras do as well. A special case of the Yangian parameters corresponds to the β-deformation. The preservation of Serre relations can be thought of a selection rule for proper systems of commuting β-deformed Hamiltonians. On the contrary, commutativity in the extended family associated with “rational (non-integer) rays” is not reduced to the Serre relations, and uses also other relations in the W1+∞ algebra. Thus their β-deformation is less straightforward.