ZZ-graded identities of the Lie algebra W1W1

Let K   be a field of characteristic 0 and let W1W1 be the Lie algebra of the derivations of the polynomial ring K[t]K[t]. The algebra W1W1 admits a natural ZZ-grading. We describe the graded identities of W1W1 for this grading. It turns out that all these ZZ-graded identities are consequences of a collection of polynomials of degree 1, 2 and 3 and that they do not admit a finite basis. Recall that the “ordinary” (non-graded) identities of W1W1 coincide with the identities of the Lie algebra of the vector fields on the line and it is a long-standing open problem to find a basis for these identities. We hope that our paper might be a step to solving this problem.