Boundary-Localized Commutators and Cohomology of Shift Algebras on the Half-Lattice: Structure, Representations, and Extensions

We study the boundary-localized Lie algebra generated by the rank-one perturbation $T=U+\epsilon E$ of the unilateral shift on ${\mathrm{\ell }}^2({\mathbb{Z}}_{\ge 0})$. While the polynomial algebra $⟨T⟩$ is abelian, the enlarged algebra $\mathcal{A}=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{U^{a}E{U}^b,U^n\}$ exhibits finite–rank commutators confined to a finite neighborhood of the boundary. We construct explicit site-localized 2-cocycles ${\omega }_j(X,Y)=⟨e_j,[X,Y]e_j⟩$ and prove they form a basis of $H^2(\mathcal{A},\mathbb{C})$. Quantitative bounds and finite-dimensional models confirm a sharp bulk-edge dichotomy. The framework provides a rigorous Lie-algebraic model for edge phenomena in discrete quantum systems-without violating the Jacobi identity.