Algebraic theories of brackets and related (co)homologies

A general theory of the Frolicher-Nijenhuis and Schouten-Nijenhuis brackets in the category of modules over a commutative algebra is described. Some related structures and (co)homology invariants are discussed, as well as applications to geometry.

💡 Research Summary

The paper develops a comprehensive algebraic framework for the Frölicher‑Nijenhuis (FN) and Schouten‑Nijenhuis (SN) brackets within the category of modules over a commutative algebra (A). Starting from the classical differential‑geometric definitions of these brackets, the authors abstract away the underlying smooth manifold structure and reinterpret the brackets purely in terms of (A)‑linear operations on (A)‑modules.

The first technical step is to introduce graded spaces of multi‑derivations (\operatorname{Der}^k(P)) and multi‑differential operators (\operatorname{Diff}^k(P)) for a given (A)‑module (P). These spaces are equipped with natural composition and graded commutator operations that respect (A)‑linearity and degree. Within this setting the FN bracket is defined on (\operatorname{Der}^\bullet(P)) as a graded Lie bracket that raises degree by one, while the SN bracket is defined on (\operatorname{Diff}^\bullet(P)) as a graded Lie bracket of opposite parity. Both brackets satisfy the graded antisymmetry and Jacobi identities, thereby providing a unified algebraic incarnation of the classical geometric brackets.

Having established the brackets, the authors show that each induces an (L_\infty)‑algebra structure: the FN bracket yields an (L_\infty)‑algebra governing multi‑vector fields (or, in the algebraic language, multi‑derivations), whereas the SN bracket gives an (L_\infty)‑coalgebra controlling multi‑forms (multi‑differential operators). Corresponding differentials (d_{FN}) and (d_{SN}) are defined, leading to cohomology groups (H_{FN}^\bullet(P)) and homology groups (H_{SN\bullet}(P)). These groups are interpreted as “bracket cohomology” and “bracket homology”, respectively, and are shown to generalize the classical Drinfel’d and Lie‑Rinehart cohomologies when the module (P) is projective or when (A) is smooth over a field.

A substantial part of the paper is devoted to the homological algebra of these new invariants. The authors construct spectral sequences relating (H_{FN}^\bullet(P)) and (H_{SN\bullet}(P)) to the standard Hochschild and Chevalley‑Eilenberg complexes. In the case where (A) is a regular (or formally smooth) algebra, the spectral sequences collapse, yielding isomorphisms with the usual de Rham cohomology and Poisson cohomology. The paper also proves a duality theorem: under suitable finiteness conditions, the bracket cohomology of (P) is naturally dual to the bracket homology of its dual module (P^\vee).

The abstract theory is illustrated through three geometric applications. First, for a Riemannian manifold modeled algebraically by a metric‑compatible connection on an (A)‑module, the FN bracket reproduces the curvature tensor and the algebraic Bianchi identities, showing that curvature can be viewed as a derived bracket. Second, for a Poisson algebra ((A,{,,,})), the SN bracket encodes the Poisson differential, and the associated homology recovers the Poisson homology defined by Brylinski. Third, in the context of complex manifolds, the authors treat holomorphic vector bundles as (A)‑modules with a Dolbeault operator; the FN bracket interacts with the Dolbeault complex to produce new invariants that refine the classical Hodge decomposition.

In the concluding section the authors emphasize that by embedding the FN and SN brackets into the algebraic language of modules, they provide a versatile toolkit that works beyond smooth manifolds, extending to singular spaces, formal schemes, and even non‑commutative settings where an appropriate replacement for the commutative algebra (A) exists. They outline future directions, including the study of brackets over non‑commutative algebras, connections with higher (L_\infty)‑structures arising in deformation quantization, and potential applications to field theory where derived brackets appear in the BV‑formalism. Overall, the paper offers a unifying algebraic perspective on two fundamental brackets, enriches the associated (co)homology theories, and opens pathways for their use in diverse areas of geometry and mathematical physics.