  We show how general principles of symmetry in quantum mechanics lead to
twisted notions of a group representation. This framework generalizes both the
classical 3-fold way of real/complex/quaternionic representations as well as a
corresponding 10-fold way which has appeared in condensed matter and nuclear
physics. We establish a foundation for discussing continuous families of
quantum systems. Having done so, topological phases of quantum systems can be
defined as deformation classes of continuous families of gapped Hamiltonians.
For free particles there is an additional algebraic structure on the
deformation classes leading naturally to notions of twisted equivariant
K-theory. In systems with a lattice of translational symmetries we show that
there is a canonical twisting of the equivariant K-theory of the Brillouin
torus. We give precise mathematical definitions of two invariants of the
topological phases which have played an important role in the study of
topological insulators. Twisted equivariant K-theory provides a finer
classification of topological insulators than has been previously available.
