of our classification and then provide the mathematical background needed for the analysis. In Sec. \[sec:exam\] we give some pedagogical examples for calculating the topological classification of all symmetries. In Sec. \[sec:discussion\] we discuss our results in comparison with previous classification schemes and suggest possible future extensions.

[ccccccc]{} &

  ---------------
    Crystalline
   & non-spatial
    symmetries
  ---------------

& &

  -------------
   Symmetries
        +
   Hamiltonian
  -------------

& & &

  -------------
   Classifying
      space
  -------------

\
& $\Downarrow$ & & $\Downarrow$ & & & $\Uparrow$\
$\Bigg($ &

  ------------
   Algebra of
   symmetries
  ------------

& $\hookrightarrow$ &

  -------------------------
   ${\mathbb{Z}}_2$-graded
        real algebra
  -------------------------

& $\Bigg)$ & $\mapsto$ &

  ---------------
   All Hilbert-
   space actions
  ---------------

\
\
& $B^0$ & & $B$ & & &

Summary of Results {#sec:results}
==================

$$\begin{array}{l||c||l||l||c|c|c|c}
    q & \mathrm{Extension}                      & \mathrm{Classifying~space}                & \mathrm{AZ~class} & d=0   & d=1   & d=2   & d=3
    \\ \hline\hline
    0 & {\mathbb{C}}\hookrightarrow{\mathbb{C}}\oplus{\mathbb{C}}& {\mathcal{C}}_0=\prod_{k+m=n} U(n)/(U(k)\times U(m))    & \mathrm{A}    & {\mathbb{Z}}& 0     & {\mathbb{Z}}& 0                             
    \\ \hline
    1 & {\mathbb{C}}\oplus{\mathbb{C}}\hookrightarrow M_2({\mathbb{C}})    & {\mathcal{C}}_1=U(n)                                    & \mathrm{AIII} & 0     & {\mathbb{Z}}& 0     & {\mathbb{Z}}\\ \hline\hline
    0 & {\mathbb{R}}\hookrightarrow{\mathbb{R}}\oplus{\mathbb{R}}& {\mathcal{R}}_0=\prod_{k+m=n} O(n)/(O(k)\times O(m))    & \mathrm{AI}   & {\mathbb{Z}}& 0     & 0     & 0                     
    \\ \hline
    1 & {\mathbb{R}}\oplus{\mathbb{R}}\hookrightarrow M_2({\mathbb{R}})    & {\mathcal{R}}_1=O(n)                                    & \mathrm{BDI}  & {\mathbb{Z}}_2 & {\mathbb{Z}}& 0     