Hom-unitality and hom-associative structures
Germán García Butenegro, Abdennour Kitouni, Sergei Silvestrov
TL;DR
The paper addresses how unitality interacts with hom‑associativity in general algebras by showing the twisting map must act as a left or right multiplication by a unity‑like element, and by developing one‑sided and two‑sided frameworks (including unitalization) to classify hom‑unities. It provides explicit characterizations of compatible maps via centers, nuclei, and annihilators, and constructs concrete subalgebras such as $AC(A)$, $AC_l(A)$, and $AC_r(A)$ that encode all hom‑associative structures for a given algebra. The work yields bijections between twists and multiplication operators, derives multiplicativity criteria tied to idempotents, and applies the framework to important classes like Cayley–Dickson algebras and (hom‑)Leibniz structures. These results deepen the understanding of how multiplicativity and unitality shape hom‑algebra structures and furnish tools for constructing or ruling out hom‑associative deformations. The unitalization approach further extends the theory to non‑unital algebras, revealing how hidden substructures govern possible hom‑unities and offering pathways to study representations and cohomology in this twisted setting.
Abstract
We study hom-associative structures on general, possibly non-associative algebras, focusing on one-sided and two-sided unital algebras. New characterizations and aspects of these structures, along with some important subclasses, are explored for non-associative algebras. By exploiting the observation that the twisting linear map in the hom-associativity axiom of one-sided unital hom-associative algebras is a left or right multiplication operator by an element of the algebra (obtained by the action of the twisting map on a corresponding one-sided unity), a new characterization of the multiplicative twisting operators (or, in other words, the multiplicative hom-associative algebras) is established for one-sided unital algebras. This demonstrates a strong connection between multiplicativity in hom-associative structures and the idempotents of the algebra, thereby further enhancing our understanding of the structure and special nature of multiplicative hom-associative algebra structures as a special subclass of arbitrary hom-associative structures with arbitrary linear twisting maps. Furthermore, new insights into subspaces and a subalgebra of hom-unities, algebra elements that induce hom-associativity by multiplication, are obtained. Moreover, since non-unital hom-associative algebras need not be twisted by a multiplication operator, a unitalization process is employed to describe a subalgebra of two-sided hom-unities that induce hom-associative structures on such algebras. This is formulated in terms of eigenspaces of multiplication operators within the algebra. Additionally, the obtained insights and general results about the structure and characterization of hom-algebra structures are applied to some important known general classes of non-associative algebras, such as commutative, possibly non-associative algebras, Cayley-Dickson algebras, Leibniz algebras, and hom-Leibniz algebras.