Distinct Floquet topological classifications from color-decorated frequency lattices with space-time symmetries

We consider nontrivial topological phases in Floquet systems using unitary loops and stroboscopic evolutions under a static Floquet Hamiltonian $H_F$ in the presence of dynamical space-time symmetries $G$. While the latter has been subject of out-of-equilibrium classifications that extend the ten-fold way and systems with additional crystalline symmetries to periodically driven systems, we explore the anomalous topological zero modes that arise in $H_F$ from the coexistence of a dynamical space-time symmetry $M$ and antisymmetry $A$ of $G$, and classify them using a frequency-domain formulation. Moreover, we provide an interpretation of the resulting Floquet topological phases using a frequency lattice with a decoration represented by color degrees of freedom on the lattice vertices. These colors correspond to the coefficient $N$ of the group extension $\tilde{G}$ of $G$ along the frequency lattice, given by $N=Z\rtimes H^1[A,M]$. The distinct topological classifications that arise at different energy gaps in its quasi-energy spectrum are described by the torsion product of the cohomology group $H^{2}[G,N]$ classifying the group extension.