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

Abstract: We study the boundary-localized Lie algebra generated by the rank-one perturbation (T = U + \varepsilon E) of the unilateral shift on (\ell^2(\mathbb{Z}_{\ge\ 0})). While the polynomial algebra (\langle T \rangle) is abelian, the enlarged algebra (\mathcal{A} = \mathrm{span}{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) = \langle e_j, [X,Y] e_j \rangle) 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.