A weighted regular jet algebra

Assign weight one to every $Z_g$ and weight two to every $M_a$. For a regular polynomial $F$, let ${\mathrm{deg}}_{\mathrm{w}}F$ be its weighted degree. Define

The coefficients may be formal in $\hslash$. This is a regular comparison algebra, not admission of a singular or noncompact coefficient.

The contraction in (eq:star) lowers weighted field degree by two. The operation ${\mathcal{V}}_{\xi }$ lowers weighted degree by at least one and does not lower the explicit power of $s$. Thus the product, operation, and quotient in (eq:Rees) are well defined.