Commutative subalgebras from Serre relations

TL;DR

This paper shows that Serre relations are sufficient for the commutativity of certain subalgebras in the $W_{1+ olinebreak ext{+} olinebreak ext{1}}$ algebra and its deformation to the affine Yangian, impacting integrable systems and $eta$-deformations.

Contribution

It establishes that Serre relations alone ensure commutativity in specific subalgebras of $W_{1+ olinebreak ext{+} olinebreak ext{1}}$ and affine Yangian, clarifying their role in integrable systems.

Findings

Serre relations suffice for commutativity in many subalgebras.

Serre relations are preserved under the affine Yangian deformation.

Commutativity for certain rational rays involves additional algebra 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.