Tensor product modules over the planar Galilean conformal algebra from free modules of rank one

Abstract: In this paper, we investigate the irreducible tensor product modules over the planar Galilean conformal algebra $\mathcal{G}$ named by Aizawa, which is the infinite-dimensional Galilean conformal algebra introduced by Bagchi-Gopakumar in $(2+1)$ dimensional space-time. We give the necessary and sufficient conditions for the tensor product modules of any two of $\mathcal{U}(\mathfrak{h})$-free modules of rank one over $\mathcal{G}$ to be irreducible, where $\mathfrak{h}$ is the Cartan subalgebra of $\mathcal{G}$.Furthermore, the isomorphism classes of these irreducible tensor product modules are determined. As an application, we obtain the necessary conditions for the tensor product modules of any two of $\mathcal{U}(\mathbb{C} L_0)$-free modules of rank one over Witt algebra and Heisenberg-Virasoro algebra to be irreducible.