Decomposition Theory

The aim of decomposition theory is to express a complex structure as a superposition of simpler components. There is no general rule for what is meant by simpler component and this is determined by the particular application. In an algebraic setting it is usual to examine two complementary forms of decomposition, the decomposition of states and the decomposition of representations. In this chapter we principally describe the theory relating to states, but the intimate connection between states and representations allows us to develop and exploit properties of the representations.