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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. 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.

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rdf:type	yago:WikicatTopologicalGroups yago:Abstraction100002137 yago:Group100031264 yago:WikicatPermutationGroups
rdfs:label	System of imprimitivity (en)
rdfs:comment	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. 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. (en)
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dcterms:subject	Topological groups Unitary representation theory Functional analysis Permutation groups
Wikipage page ID	1388070 (xsd:integer)
Wikipage revision ID	1095054021 (xsd:integer)
Link from a Wikipage to another Wikipage	Canonical quantization Vector space Coset Mathematical analysis Mathematics Normal subgroup George Mackey Configuration space (physics) Theory Lorentz group Identity element Permutation representation Projection-valued measure Maximal subgroup Topological groups Undefined (mathematics) Unitary representation theory Heisenberg group Linear span Locally compact group Trivial representation Algebra Ergodic Eugene Wigner Primitive permutation group Regular representation Group action (mathematics) Hilbert space Functional analysis Permutation groups Character theory Group representation Group ring If and only if Induced representation Semi-direct product Strong operator topology Unitary representation Poincaré group Linearly independent Finite group Transitive action Direct sum of vector spaces Projection operator
sameAs	System of imprimitivity System of imprimitivity System of imprimitivity System of imprimitivity
thumbnail	wiki-commons:Special:FilePath/OrbitStructureDual.png?width=300
has abstract	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 (see primitive permutation group). 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: first in terms of functions on G constant on K-cosets, and then in terms of projection operators (for example the averaging over K-cosets of elements of the group algebra). 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. (en)
prov:wasDerivedFrom	wikipedia-en:System_of_imprimitivity?oldid=1095054021&ns=0
page length (characters) of wiki page	17973 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:System_of_imprimitivity
is Link from a Wikipage to another Wikipage of	Mackey theory Wigner's classification George Mackey Glossary of representation theory Direct integral Induced representation Imprimitivity
is Wikipage redirect of	Mackey theory Imprimitivity
is foaf:primaryTopic of	wikipedia-en:System_of_imprimitivity

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