08. Scientific Journals

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Quasi-Associative Algebras
(Suleyman Demirel University, 2007) Askar Dzhumadildaev
This paper investigates a class of algebras obtained from associative algebras by introducing a q-commutator defined as a * b = ab + qba, where q belongs to a field K with characteristic not equal to 2 or 3, and q² is not equal to 0 or 1. Using this commutator, we construct algebras that form a variety characterized by a q-associativity identity. When the parameter satisfies q² − 4q + 1 ≠ 0, this identity alone describes the entire variety of q-associative algebras. However, when q² − 4q + 1 = 0, the q-associativity identity must be supplemented by the Lie-admissibility identity. In this exceptional case, the resulting variety is equivalent to the class of alternative algebras. Therefore, q-associative algebras form a structural link between associative, Lie-admissible, and alternative algebras, demonstrating how small changes in the commutator parameter q can transform fundamental algebraic properties.