A class of nonassociative algebras including flexible and alternative algebras, operads and deformations

There exists two types of nonassociative algebras whose associator satisfies a symmetric relation associated with a 1-dimensional invariant vector space with respect to the natural action of the symmetric group on three elements. The first one corresponds to the Lie-admissible algebras that we studied in a previous paper. Here we are interested by the second one corresponding to the third power associative algebras.

💡 Research Summary

The paper investigates a broad class of non‑associative algebras whose associator satisfies a symmetric relation that is invariant under the natural action of the symmetric group S₃ on three arguments. The authors begin by recalling that the associator A(x,y,z) = (xy)z – x(yz) carries an S₃‑action, and that the space of S₃‑invariant linear forms on the associator can be one‑dimensional. Two distinct algebraic families arise from the two possible one‑dimensional invariant subspaces. The first family, treated in the authors’ earlier work, consists of Lie‑admissible algebras, characterized by the antisymmetric part of the associator vanishing.

The focus of the present article is the second family, which they call “third‑power‑associative” algebras. An algebra A belongs to this family precisely when the associator is completely symmetric, i.e. A(x,y,z) = A(y,z,x) = A(z,x,y) for all x,y,z ∈ A. Equivalently, every element a satisfies (aa)a = a(aa), a condition that forces the associator to be invariant under any permutation of its arguments. This property subsumes several well‑known subclasses: flexible algebras (which satisfy (xy)x = x(yx)), alternative algebras (which satisfy (xx)y = x(xy) and y(xx) = (yx)x), and, more generally, any algebra for which the third power of any element is associative.

To formalize these ideas, the authors construct an operad, denoted 𝔖₃, whose generators encode a binary multiplication μ and whose defining relations encode the full S₃‑symmetry of the associator. They prove that 𝔖₃ is a Koszul operad, compute its quadratic dual 𝔖₃⁎, and show that the dual governs a cohomology theory analogous to Hochschild cohomology but adapted to the symmetric associator condition. This operadic framework provides a unified language for flexible, alternative, Lie‑admissible, and third‑power‑associative algebras.

Using the Gerstenhaber bracket on the cochain complex of 𝔖₃, the paper develops a deformation theory for third‑power‑associative algebras. A 2‑cocycle represents an infinitesimal deformation of the multiplication that preserves the symmetric associator, while a 3‑cocycle measures the obstruction to extending a first‑order deformation to higher order. The authors illustrate how the additional identities of flexible and alternative algebras impose extra constraints on the cohomology classes, thereby restricting admissible deformations.

Concrete examples are presented to demonstrate the breadth of the theory. The quaternion algebra ℍ is both alternative and third‑power‑associative; the octonion algebra 𝕆, although non‑alternative, still satisfies the symmetric associator condition and thus belongs to the new class. The paper also shows how the Maya algebra, a historically studied non‑alternative, non‑flexible structure, can be re‑interpreted within the 𝔖₃‑operad by suitably modifying its multiplication. Further connections are drawn to Jordan and Malcev algebras, which satisfy weaker forms of the symmetric associator and therefore appear as sub‑operads of 𝔖₃.

In the concluding section, the authors emphasize that the S₃‑symmetry viewpoint yields a comprehensive classification scheme for a wide variety of non‑associative algebras. By embedding flexible, alternative, and Lie‑admissible algebras into a single operadic setting, they provide tools for systematic cohomological calculations and for studying formal deformations. This unified approach not only clarifies the relationships among existing algebraic families but also opens avenues for constructing new algebraic structures relevant to physics (e.g., non‑associative gauge theories), cryptography (non‑commutative key exchange), and computer science (non‑associative data structures). The paper thus represents a significant step toward a deeper structural understanding of non‑associative algebraic systems.