Algebraic Weyl modules and exponentiated representations are different objects

The algebra $W_n(k;\mathrm{\hslash })$ is the algebraic tensor-algebra quotient defined in Polynomial Weyl algebra and its central parameter. Its definition supplies no topology, involution, Hilbert space, or one-parameter unitary groups. An algebraic module alone therefore does not specify exponentiated unitary operators.

The precise sufficient bridge proved here is A common core with graph-differentiable orbits integrates Heisenberg commutators. It assumes self-adjoint generators, a common invariant graph core, invariance under their functional-calculus unitary groups, and explicit differentiability of each unitary orbit in every generator's graph norm. The proof integrates the commutator on the core, establishes equality of the closed conjugated operators including their domains, and then obtains the Weyl relations and their finite-dimensional cocycle. Common invariant density alone is not asserted to suffice. The lemma carries the Axiom of Choice and Countable Choice used by its spectral/Stone and self-adjoint suppliers.

The later Exponentiated Weyl Relations and Regular Representations page (ID regular-weyl-representations-and-stone-von-neumann) may reuse that exact conditional bridge. Its universal $C^{\ast }$-algebra, existence and Stone–von Neumann uniqueness claims remain separate obligations; neither the algebraic normal form nor the bridge establishes those claims.