Notes on Cohomologies of Ternary Algebras of Associative Type

The aim of this paper is to investigate the cohomologies for ternary algebras of associative type. We study in particular the cases of partially associative ternary algebras and weak totally associative ternary algebras. Also, we consider the Takhtajan’s construction, which was used to construct a cohomology of ternary Nambu-Lie algebras using Chevalley-Eilenberg cohomology of Lie algebras, and discuss it in the case of ternary algebras of associative type. One of the main results of this paper states that a deformation cohomology does not exist for partially associative ternary algebras which implies that their operad is not a Koszul operad.

💡 Research Summary

The paper investigates cohomology theories for ternary algebras of associative type, focusing on two subclasses: partially associative ternary algebras and weak totally associative ternary algebras. After recalling the definition of an associative‑type ternary algebra—namely a vector space equipped with a trilinear product μ: A⊗A⊗A → A that satisfies a suitable associativity condition—the authors distinguish the two cases by the precise form of the associativity identity. In the partially associative case the identity holds only when the nesting of μ involves the first two slots, while in the weak totally associative case the identity holds for all possible nestings, i.e. μ(μ(a,b,c),d,e)=μ(a,μ(b,c,d),e)=μ(a,b,μ(c,d,e)).

To construct a deformation cohomology, the authors introduce cochain complexes Cⁿ(A,A) for n=1,2,… together with a coboundary operator δ that is defined by inserting the ternary product into cochains in all admissible positions. For binary associative algebras this reproduces the Hochschild complex; the question is whether an analogous complex can be built for ternary structures. The authors compute δ² on a generic 2‑cochain. In the partially associative situation extra terms survive, violating the fundamental requirement δ²=0. Consequently no consistent 2‑cochain complex exists, and the deformation theory collapses already at second order. This failure is interpreted operad‑theoretically: the quadratic operad governing partially associative ternary algebras is not Koszul, because the Koszul dual complex does not resolve the operad.

In contrast, for weak totally associative ternary algebras the authors succeed in defining a full cochain complex. The extra associativity constraints guarantee that all unwanted terms cancel, and δ²=0 holds. They compute the low‑dimensional cohomology groups H¹, H² and H³ for several concrete examples (e.g., ternary matrix algebras, function algebras with pointwise ternary product). H² classifies infinitesimal deformations, while H³ contains the obstruction classes that decide whether a first‑order deformation can be extended to higher order. The results show that a rich deformation theory exists for this class, mirroring the classical Hochschild theory for binary associative algebras.

The paper then revisits Takhtajan’s construction, originally devised to obtain a cohomology for ternary Nambu‑Lie algebras from the Chevalley‑Eilenberg cohomology of Lie algebras. The authors adapt the idea to associative‑type ternary algebras by first embedding the ternary product into a binary one (μ(a,b,c) =