From Derivations to Automorphism: A Cohomological Classification of n-Structures in Nest Algebras

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DOI: https://doi.org/10.28924/2291-8639-24-2026-92
Published: Mar 19, 2026
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Khalil H. Hakami, From Derivations to Automorphism: A Cohomological Classification of n-Structures in Nest Algebras, Int. J. Anal. Appl., 24 (2026), 92.

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Khalil H. Hakami

Abstract

This paper develops a unified cohomological framework for analyzing n-derivations and their induced derivational automorphisms on nest algebras, extending classical Hochschild cohomology into the realm of higher-order operator algebraic structures. By constructing explicit n-cochain complexes adapted to the triangular nature of nest algebras, we classify n-derivations up to cohomological equivalence and provide precise criteria for their innerness. We demonstrate that the vanishing of higher-order cohomology groups corresponds to structural rigidity, while non-trivial cohomology classes reflect obstructions to decomposability and inner implementation. Moreover, we show that cohomologically trivial n-derivations preserve essential algebraic features, including the radical, center, invariant subspaces, and two-sided ideals. Through explicit computations of Hn(A,A) for low-dimensional examples, we verify the existence of non-trivial cohomological classes. Additionally, we introduce a dual cohomology theory for n-automorphisms via exponential mappings and establish a bidirectional correspondence between infinitesimal derivations and global automorphisms. This approach unifies derivational and automorphic symmetries under a single cohomological classification, offering new perspectives toward deformation theory, categorical dualities, and quantum operator structures.

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