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The concept of system of imprimitivity is used in mathematics, particularly in algebra and analysis, both within the context of the theory of group representations. It was used by George Mackey as the basis for his theory of induced unitary representations of locally compact groups.
The simplest case, and the context in which the idea was first noticed, is that of finite groups. Consider a group G and subgroups H and K, with K contained in H. Then the left cosets of H in G are each the union of left cosets of K. Not only that, but translation (on one side) by any element g of G respects this decomposition. The connection with induced representations is that the permutation representation on cosets is the special case of induced representation, in which a representation is induced from a trivial representation. The structure, combinatorial in this case, respected by translation shows that either K is a maximal subgroup of G, or there is a system of imprimitivity (roughly, a lack of full 'mixing'). In order to generalise this to other cases, the concept is re-expressed in terms of projection operators.
Mackey also used the idea for his explication of quantization theory based on preservation of relativity groups acting on configuration space. This generalized work of Eugene Wigner and others and is often considered to be one of the pioneering ideas in canonical quantization.
Definitions
Mackey's original formulation was expressed in terms of a locally compact second countable (lcsc) group G, a standard Borel space X and a Borel group action
- <math> G \times X \rightarrow X, \quad (g,x) \mapsto g \cdot x. <math>
The definitions can be given in a much more general context, but the original setup used by Mackey is still quite general and requires fewer technicalities.
Definition. Let G be a lcsc group acting on a standard Borel space X. A system of imprimitivity based on (G, X) consists of a separable Hilbert space H and a pair consisting of
which satisfy
- <math> U_g \pi(A) U_{g^{-1}} = \pi(g \cdot A). <math>
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