Dellalgebras

Dellalgebras are a class of algebraic structures defined over a field, consisting of a vector space A equipped with a bilinear product ∘ and a linear operator δ: A → A, called the dellal operator. They are presented as a generalization of more familiar nonassociative systems and are studied in the context of understanding how a derivation-like operation interacts with a binary product.

The defining properties of a dellalgebra are given by two main identities that relate δ to the product. First, δ is required to act as a derivation with respect to the product, meaning δ(x ∘ y) = δ(x) ∘ y + x ∘ δ(y) for all elements x and y in A. Second, a Dellal associator condition governs the extent to which the product fails to be associative, tying the associator [x,y,z] = (x ∘ y) ∘ z − x ∘ (y ∘ z) to δ and the product in a prescribed way. Together, these identities restrict the algebra’s structure and place dellalgebras in a framework that generalizes both associative algebras and certain derivation-augmented nonassociative systems.

Key properties include stability under taking subalgebras and homomorphisms that preserve both ∘ and δ, and the existence of notions such as simple or semisimple dellalgebras. Representation theory and module categories for dellalgebras have been considered, with attention to how modules interact with the dellal operator.

Examples range from the trivial cases, such as the zero product with arbitrary δ, to associative algebras equipped with δ = 0, which satisfy the dellal identities automatically. More elaborate nonassociative instances can be constructed by choosing specific δ and product rules that satisfy the defining axioms.

Dellalgebras have appeared in theoretical discussions as a flexible generalization tool for examining the interplay between derivations and products, and they are sometimes explored for potential connections to broader topics in nonassociative algebra and mathematical physics.