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

TL;DR

This paper investigates the structure and cohomology of boundary-localized shift algebras on the half-lattice, revealing finite-rank commutators, explicit cocycles, and a bulk-edge dichotomy relevant for discrete quantum systems.

Contribution

It introduces a boundary-localized Lie algebra framework with explicit cohomology basis and models edge phenomena in quantum systems without violating Jacobi identity.

Findings

Finite-rank commutators confined near the boundary.

Explicit site-localized 2-cocycles form a basis of the second cohomology.

Quantitative bounds confirm a sharp bulk-edge dichotomy.

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.