Classification of generalized \(A\)-graded algebras with 3 generators (Q1897531)

Let \(B\) be an \(N\)-graded algebra over the field \(\mathbb{C}\) of complex numbers generated by finitely many homogeneous elements \(x_1, \dots, x_n\) of degrees \(p_1, \dots, p_n\), respectively. \(B\) is called an \(A\)-graded algebra if all the nontrivial homogeneous subspaces of \(B\) have the dimension 1. For any linear function \(\varphi : R^n \to R\) and degrees \(p_1, \dots, p_n\) the authors construct an \(A\)-graded algebra \(B\) generated by \(n\) homogeneous elements of degrees \(p_1, \dots, p_n\). Extending results of V. I. Arnold the authors characterize all \(A\)-graded algebras generated by three elements in terms of linear functions. They show also that for every triple \((p_1, p_2, p_3)\) there are only finitely many \(A\)-graded algebras generated by three homogeneous elements of degrees \(p_1, p_2, p_3\), respectively.