Quasi-Associative Algebras
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Date
2007
Authors
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Journal ISSN
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Publisher
Suleyman Demirel University
Abstract
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.
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Keywords
quasi-associative algebras, q-commutator, algebraic identities, associators, Lie-admissible algebras, alternative algebras
Citation
Askar Dzhumadildaev / Quasi-Associative Algebras / Suleyman Demirel University / Сду хабаршысы, 2007