Quantization of the theory of topological insulators

Topological insulators are the solid bodies having a broad energy gap stable under small deformations. This motivates the usage of topological methods for their study.
A key role in the theory of solid states is played by their symmetry groups. Kitaev has pointed out that there is a relation between the symmetries of solid bodies and Clifford algebras. According to this observation the quantization of topological insulators should reduce to the theory of irreducible representations of Clifford algebras.
The next important step was done by Kennedy and Zirnbauer who introduced the notion of pseudosymmetries. While the algebra of observables of a topological insulator is generated by Hamiltonians satisfying commutation relations with symmetry operators, the quantum obsrvables are given by the complex sstructures on the Nambu space satisfying anticommutation relations with pseudosymmetries. This correspondence determines the quantization of topological insulators.