Identities for subspaces of the Weyl algebra (Q6591996)

Let \(\mathbb{F}\) be an infinite field of arbitrary characteristic and let \(\textsf{A}_1\) be the Weyl algebra over \(\mathbb{F}\) in one variable, generated by \(x\) and \(\partial_x\), that can be seen as a Lie algebra with the usual bracket given by the commutator. Let \(\textsf{A}_1^{(-,1)}\) be the Lie subalgebra of \(\textsf{A}_1\) generated by the \(\mathbb{F}\)-linear span of the \(x^k\partial_x\), for all \(k\geq 0\). This Lie algebra is isomorphic to the Witt algebra, which is of interest because it is simple and infinite dimensional. In the paper under review, the authors provide a basis of the vector spaces of polynomial identities for \(\textsf{A}_1^{(-,1)}\) of degree 4, distinguishing four different cases attending to their multidegrees.