Theorem: Restricted classical nilpotent continuation

The domain ${\mathcal{B}}_{{\mathcal{R}}_{\mathrm{cl}}}$:

1. is a graded subalgebra for the classical product and the odd bracket;

2. is preserved by $\mathcal{Q}$;

3. carries the nilpotent differential $\mathcal{Q}{|}_{{\mathcal{B}}_{{\mathcal{R}}_{\mathrm{cl}}}}$;

4. contains the complete field-only classical algebra, on which $\mathcal{Q}=s$;

5. is the largest subdomain of $\mathcal{B}$ on which the same Hamiltonian operator has square zero.

This construction does not set ${\mathcal{R}}_{\mathrm{cl}}=0$ in the ambient algebra.

Proof. The map $({\mathcal{R}}_{\mathrm{cl}},\cdot )$ is an even derivation of both the product and bracket. Its kernel is therefore a graded subalgebra for both operations.

By (eq:curvature) and Jacobi,

Thus $({\mathcal{R}}_{\mathrm{cl}},F)=0$ implies $({\mathcal{R}}_{\mathrm{cl}},\mathcal{Q}F)=0$. Equation (eq:curvature) then proves nilpotence.

Both ${\mathcal{R}}_{\mathrm{cl}}$ and a field-only functional depend on no antifields, so their antibracket is zero. On such functionals $\mathcal{Q}=s$.

Finally, if the same $\mathcal{Q}$ has square zero on an element $F$, then (eq:curvature) forces $({\mathcal{R}}_{\mathrm{cl}},F)=0$. Every square-zero domain is contained in (eq:centralizer).