Korean J. Math.  Vol 26, No 2 (2018)  pp.175-189
DOI: https://doi.org/10.11568/kjm.2018.26.2.175

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

A.H. Rezaei, Bijan Davvaz


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.


$\Gamma$-algebra, $\alpha$-Lie algebra, $\alpha$-ideal, $\alpha$-derivation, $\alpha$-representation, $\alpha$-nilpotent.

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