\(\mathbb Z^ 2\)-graded modules with one-dimensional components over the Lie algebra \(A_ 1^{(1)}\) (Q1092999)

The infinite-dimensional Kac-Moody algebras have the representations with highest weights. These representations are of exponential growth with respect to natural \(\mathbb Z^ r\)-graduation where \(r\) is the rank of the algebra. In this short paper the author gives the classification of those \(\mathbb Z^ 2\)-graded representations of the Kac-Moody algebra \(A_ 1^{(1)}\) which have one-dimensional homogeneous components. The basis and the formulas of the action of elements of the algebra onto basis elements are given for each representation. A generalization of these representations is discussed.