Difference between revisions of "Metaplectic group:Mp(2,R)"

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(Created page with "{{particular group}} ==Definition== This group can be defined in the following equivalent ways: # It is the metaplectic group <math>Mp(2,\R)</math>, i.e., it has degree...")
 
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# It is the [[metaplectic group]] <math>Mp(2,\R)</math>, i.e., it has degree two over the [[field of real numbers]].
 
# It is the [[metaplectic group]] <math>Mp(2,\R)</math>, i.e., it has degree two over the [[field of real numbers]].
 
# It is the unique double cover of [[special linear group:SL(2,R)]].
 
# It is the unique double cover of [[special linear group:SL(2,R)]].
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=Arithmetic functions==
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{| class="sortable" border="1"
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! Function !! Value !! Similar groups !! Explanation
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|-
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| [[order of a group]] || cardinality of the continuum || || Same infinite cardinality as [[SL(2,R)]].
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|-
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| [[exponent of a group]] || infinite || || Same as [[SL(2,R)]].
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|-
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| {{arithmetic function value|composition length|3}} || || We can construct a composition series that begins with a subgroup of order two for which the quotient is [[SL(2,R)]], then proceeds to the inverse image of the center of <math>SL(2,\R)</math> in <math>Mp(2,\R)</math>, and then to the whole group. The successive quotients are [[cyclic group:Z2]], [[cyclic group:Z2]], and [[PSL(2,R)]]. See also [[projective special linear group is simple]].
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|-
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| {{arithmetic function value|chief length|3}} || || The unique chief series is the same as the composition series above.
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| {{arithmetic function value|dimension of a real Lie group|3}} || || As <math>Mp(n,\R), n = 2: n(n+1)/2 = 2(2 + 1)/2 = 3</math><br>As double cover of <math>SL(2,\R)</math>: Same as dimension of <math>SL(2,\R)</math>, which is 3.
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Revision as of 16:21, 18 September 2012

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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Definition

This group can be defined in the following equivalent ways:

  1. It is the metaplectic group Mp(2,\R), i.e., it has degree two over the field of real numbers.
  2. It is the unique double cover of special linear group:SL(2,R).

Arithmetic functions=

Function Value Similar groups Explanation
order of a group cardinality of the continuum Same infinite cardinality as SL(2,R).
exponent of a group infinite Same as SL(2,R).
composition length 3 We can construct a composition series that begins with a subgroup of order two for which the quotient is SL(2,R), then proceeds to the inverse image of the center of SL(2,\R) in Mp(2,\R), and then to the whole group. The successive quotients are cyclic group:Z2, cyclic group:Z2, and PSL(2,R). See also projective special linear group is simple.
chief length 3 The unique chief series is the same as the composition series above.
dimension of a real Lie group 3 As Mp(n,\R), n = 2: n(n+1)/2 = 2(2 + 1)/2 = 3
As double cover of SL(2,\R): Same as dimension of SL(2,\R), which is 3.