Projective geometries, $Q$-polynomial structures, and quantum groups

In 2023 we obtained a $Q$-polynomial structure for the projective geometry $L_N(q)$. In the present paper, we display a more general $Q$-polynomial structure for $L_N(q)$. Our new $Q$-polynomial structure is defined using a free parameter $\varphi$ that takes any positive real value. For $\varphi=1$ we recover the original $Q$-polynomial structure. We interpret the new $Q$-polynomial structure using the quantum group $U_{q^{1/2}}(\mathfrak{sl}_2)$ in the equitable presentation. We use the new $Q$-polynomial structure to obtain analogs of the four split decompositions that appear in the theory of $Q$-polynomial distance-regular graphs.