Corollary: No coefficientwise quantum restriction quotient

There is no associative product on

for which coefficientwise restriction is a unital homomorphism from the inherited ${\star }_H$-algebra.

More generally, a unital $\mathbb{C}[[\hslash ,\kappa ]]$-algebra homomorphism which kills every $L_q$ while preserving the inherited brackets must satisfy $\hslash 1=0$ in its target. It cannot have a nonzero unit with faithful formal scalar inclusion, or be a nonzero torsion-free formal quantum algebra.

Proof. The kernel of a homomorphism is a two-sided ideal, contradicting Theorem thm:nonideal. Alternatively, apply a proposed homomorphism to (eq:commutator). Its left side is zero because the image of $L_q$ is zero. Since $C(q,h)$ is a nonzero scalar, the result forces $\hslash 1=0$. In the coefficientwise restriction module the class of $1$ is nonzero, and multiplication by $\hslash$ merely shifts coefficients, so this relation is impossible.