Projective geometries, Q-polynomial structures, and quantum groups

In 2023 we obtained a Q-polynomial structure for the projective geometry $L_N\left(q\right)$. In the present paper, we display a more general Q-polynomial structure for $L_N\left(q\right)$. Our new Q-polynomial structure is defined using a free parameter φ that takes any positive real value. For $\phi =1$ we recover the original Q-polynomial structure. We interpret the new Q-polynomial structure using the quantum group $U_{q^{1/2}}\left({\mathrm{sl}}_2\right)$ 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.