Boundary-Localized Commutators and Cohomology of Shift Algebras on the Half-Lattice: Structure, Representations, and Extensions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911251044499456 |
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| author | Athmouni, Nassim |
| author_facet | Athmouni, Nassim |
| contents | 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 \(ω_j(X,Y) = \langle e_j, [X,Y] e_j \rangle\) 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_03927 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Boundary-Localized Commutators and Cohomology of Shift Algebras on the Half-Lattice: Structure, Representations, and Extensions Athmouni, Nassim Rings and Algebras Functional Analysis Operator Algebras Representation Theory 47B35, 47L60, 17B65, 17B56, 47A53 G.1.1; I.1.2; F.2.2 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 \(ω_j(X,Y) = \langle e_j, [X,Y] e_j \rangle\) 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. |
| title | Boundary-Localized Commutators and Cohomology of Shift Algebras on the Half-Lattice: Structure, Representations, and Extensions |
| topic | Rings and Algebras Functional Analysis Operator Algebras Representation Theory 47B35, 47L60, 17B65, 17B56, 47A53 G.1.1; I.1.2; F.2.2 |
| url | https://arxiv.org/abs/2511.03927 |