Stability and commutation relations for quantum Volterra hierarchies

Let $\mathcal J$ and $\hat{\mathcal J}$ be the ideals defining the quotient algebras $\mathcal A_{\mathcal J}$ and $\mathcal A_{\hat{\mathcal J}}$, and let $\partial_{t_\ell}$, $\partial_{\tau_m}$, $H_\ell$, $\hat H_{2\ell}$, $K^{(\ell)}$, and $K^{(\ell)+}$ denote the flows, Hamiltonians, and associated quantities of the Volterra hierarchies. For all $\ell,m\in\mathbb N$, stability and commutation conjecture. (i) The ideal $\mathcal J$ is $\partial_{t_\ell}$-stable, and in $\mathcal A_{\mathcal J}$ one has

$$[\partial_{t_\ell},\partial_{\tau_m}]=0,$$

and

$$[H_\ell,u]=K^{(\ell)}-K^{(\ell)+}.$$

(ii) The ideal $\hat{\mathcal J}$ is $\partial_{t_{2\ell}}$-stable, and in $\mathcal A_{\hat{\mathcal J}}$ one has

$$[\partial_{t_{2\ell}},\partial_{\tau_{2m}}]=0,$$

and

$$[\hat H_{2\ell},u]=K^{(2\ell)}-K^{(2\ell)+}.$$

These identities describe the compatibility of the Volterra hierarchy flows with the quotient algebras associated with the standard and non-standard quantum constructions. The supplied text does not identify this statement as conjectural or provide evidence resolving it, so its status is left open.