Non-weight modules over the algebra \(\cal{SW}(b)\)

In this paper, we denote the semi-direct product of the Witt algebra and the loop Schrödinger algebra by \(\cal{SW}(b)\) $$\mathcal{S}\mathcal{W}(b)$$ , where b belongs to ℂ. Our primary focus is on classifying U(ℂd₀ ⊕ ℂh₀)-free modules of rank 1 over \(\cal{SW}(b)\) $$\mathcal{S}\mathcal{W}(b)$$ . We characterize both the irreducibility and isomorphism classes of these modules. Furthermore, we construct new non-weight modules over \(\cal{SW}(0)\) $$\mathcal{S}\mathcal{W}(0)$$ by taking the tensor product of U(ℂd₀ ⊕ ℂh₀)-free modules with irreducible highest weight modules. We also consider the irreducibility and isomorphism classes for the tensor product modules. Finally, we reformulate some tensor product modules over \(\cal{SW}(0)\) $$\mathcal{S}\mathcal{W}(0)$$ as induced modules derived from modules over certain subalgebras.