Periodic Algebras Generated by Groups

We consider algebras with basis numerated by elements of a group G. We fix a function f from G x G to a ground field and give a multiplication of the algebra which depends on f. We study the basic properties of such algebras. In particular, we find a condition on f under which the corresponding algebra is a Leibniz algebra. Moreover, for a given subgroup (G) over cap of G we define a (G) over cap -periodic algebra, which corresponds to a (G) over cap -periodic function f, we establish a criterion for the right nilpotency of a (G) over cap -periodic algebra. In addition, for G = Z we describe all 2Z- and 3Z-periodic algebras. Some properties of nZ-periodic algebras are obtained.