Archive | 2021
Quasi-Associative Algebras on the Frobenius Lie Algebra M_3 (R)⊕gl_3 (R)
Abstract
In this paper, we study the quasi-associative algebra property for the real Frobenius\xa0 Lie algebra \xa0of dimension 18. The work aims\xa0 to prove that \xa0is a quasi-associative algebra and to compute its formulas explicitly. To achieve this aim, we apply the literature reviews method corresponding to Frobenius Lie algebras, Frobenius functionals, and the structures of quasi-associative algebras. In the first step, we choose a Frobenius functional determined by direct computations of a bracket matrix of \xa0and in the second step, using an induced symplectic structure, we obtain the explicit formulas of quasi-associative algebras for . As the results, we proved that \xa0has the quasi-associative algebras property, and we gave their formulas explicitly. For future research, the case of the quasi-associative algebras on\xa0 \xa0is still an open problem to be investigated. Our result can motivate to solve this problem.