Irreducible tensor product modules over the W-algebra W(2,2)

In this paper, we construct a class of non-weight modules [Formula: see text] over the [Formula: see text]-algebra [Formula: see text] by taking tensor products of modules [Formula: see text] with restricted modules [Formula: see text]. Then, we give the necessary and sufficient conditions for the irreducibility of such tensor product modules and determine the conditions under which two irreducible tensor product modules are isomorphic. These non-weight modules are different from known non-weight modules. Finally, we transform some tensor product modules into induced modules from modules of certain subalgebras over the [Formula: see text]-algebra [Formula: see text]. And the conditions for these induced modules to be irreducible are determined by applying the established results.