Construction of $\Gamma$-algebra and $\Gamma$-Lie admissible algebras

Abstract

In this paper,  at first we generalize the notion of algebra over a field. A $\Gamma$-algebra is an algebraic structure consisting of a vector space $V$, a groupoid $\Gamma$ together with a map from $V\times\Gamma\times V$ to $V$. Then, on every associative $\Gamma$-algebra $V$  and for every $\alpha\in \Gamma$ we construct an $\alpha$-Lie algebra. Also, we discuss some properties about $\Gamma$-Lie algebras when $V$ and $\Gamma$ are the sets of $m\times n$ and $n\times m$ matrices over a field $F$ respectively. Finally, we define the notions of $\alpha$-derivation, $\alpha$-representation, $\alpha$-nilpotency and prove Engel theorem in this case.