# Semi-Associative $3$-Algebras

@article{Bai2019SemiAssociative, title={Semi-Associative \$3\$-Algebras}, author={Ruipu Bai and Yan Zhang}, journal={arXiv: Mathematical Physics}, year={2019} }

A new 3-ary non-associative algebra, which is called a semi-associative $3$-algebra, is introduced, and the double modules and double extensions by cocycles are provided. Every semi-associative $3$-algebra $(A, \{ , , \})$ has an adjacent 3-Lie algebra $(A, [ , , ]_c)$. From a semi-associative $3$-algebra $(A, \{, , \})$, a double module $(\phi, \psi, M)$ and a cocycle $\theta$, a semi-direct product semi-associative $3$-algebra $A\ltimes_{\phi\psi} M $ and a double extension $(A\dot+A…

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