Semi-Associative $3$-Algebras

Abstract

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 $(\varphi ,\psi ,M)$ and a cocycle $\theta$, a semi-direct product semi-associative $3$-algebra $A{⋉}_{\varphi \psi }M$ and a double extension $(A\dot{+}{A}^{\ast },\{,,{\}}_{\theta })$ are constructed, and structures are studied.