Dihedral invariants of relatively free metabelian algebras

Let F be the free metabelian Leibniz algebra over with free generators x1 and x2; here metabelian means that the derived algebra [F, F] is abelian. We also consider the free metabelian bicommutative algebra M over with free generators x1 and x2. In the present study, we obtain a free generating set of [F, F]D2n as a module, where D2n is the dihedral group of order 2n. Next, we show that the algebra MD2n of D2n-invariants is three generated as a -module.