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Classification of generalized A-graded algebras with 3 generators

We study N-graded algebras, which are a quotient of the polynomial algebra C [x 1 ,..., x n ], such that the non-trivial homogeneous subspaces have dimension 1. We call these algebras A-graded algebras. We prove that to any linear functional φ: R n → R one can associate such an algebra. Moreover, we show that in case n = 3 all A-graded algebras are obtained in this way. This leads to a full classification of these algebras for n = 3. There turn out to be only a finite number of non-isomorphic A-graded algebras, once the degrees of x 1 , x 2 and x 3 are fixed