CONSTRUCTION OF Gamma-ALGEBRA AND Gamma-LIE ADMISSIBLE ALGEBRAS

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 together with a map from V x Gamma x V to V. Then, on every associative Gamma-algebra V and for every alpha is an element of 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 x n and n x 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.