Derivations and Representations of a Class of Algebras Satisfying an Osborn Identity
摘要
New classes of non-associative algebras satisfying polynomial identities were introduced by Osborn in the 1960s. Their research brought novel insights in graph theory, cryptography, computer science, and physics. All degree five irreducible identities that do not result from commutativity were found by him. This paper examines a class of non-associative commutative algebras that satisfy the identity \(\displaystyle \begin{array}{@{}rcl@{}} -y(x^2.x^2) +5(yx^3)x-9((yx^2)x)x + 4(((yx)x)x)x + ((yx)x^2)x \\ + (yx^2)x^2 - (yx)x^3 =0, \end{array} \) which is one of the Osborn names. We demonstrate that all algebras belonging to this class allow a Peirce decomposition, presuming that there is a non-zero idempotent. With this decomposition, we describe the products of the Peirce subspaces and explore the derivations relevant to algebras in this class. Finally, we determines the representations of these algebras and we highlight the irreducible submodules and elucidate their properties.