Hom-Algebra Structures and Deformation Theory

Hom-algebra theory extends classical associative, Lie and Poisson frameworks by twisting defining identities through linear self-maps. Originating from q-deformations and discretisations, Hom-algebras accommodate controlled deviations from strict associativity or the Jacobi identity. In this setting, Hom-Lie and Hom-associative structures are interwoven with cohomological methods to study infinitesimal and formal deformations. Deformation theory provides a systematic approach to perturb algebraic operations, classifying nontrivial extensions and obstructions via graded cohomology complexes. Central operators such as Nijenhuis and Rota–Baxter endomorphisms generate trivial and nontrivial deformations, while BiHom-algebras introduce pairs of commuting twists, further enriching the landscape. The resulting toolkit has found applications in integrable systems, noncommutative geometry, conformal field theory and tight-binding models in condensed matter physics. By elucidating the interplay between homomorphisms, brackets and products, current research aims both to chart new examples of Hom-algebras and to reveal their role in broader mathematical and physical contexts.

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Hom-Algebra Structures and Deformation Theory publication trend

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Technical terms

Hom-algebra: An algebraic system whose defining identities (associativity, Jacobi, etc.) are twisted by a linear self-map, generalising classical algebra structures.

Deformation theory: A framework to study continuous or formal perturbations of algebraic operations, classifying extensions and obstructions via cohomology.

Cohomology: A graded vector space of multilinear cochains with a differential that measures the failure of deformed operations to satisfy original identities.

Rota–Baxter operator: A linear endomorphism satisfying an integration-type identity that produces new algebraic deformations and Hom-structures.

BiHom-Poisson conformal algebra: A conformal algebra endowed with two commuting twist maps, intertwining conformal and Poisson operations under compatibility conditions.

References

1. Hom-Lie-Virasoro symmetries in Bloch electron systems and quantum plane in tight binding models. Nuclear Physics B (2023).
2. Constructing and Analyzing BiHom-(Pre-)Poisson Conformal Algebras. Symmetry (2024).
3. Generalized Derivations and Rota-Baxter Operators of n-ary Hom-Nambu Superalgebras. Advances in Applied Clifford Algebras (2021).