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In [[mathematics]], if {{math|''G''}} is a [[group (mathematics)|group]] and {{math|Π}} is a [[representation theory|representation]] of it over the [[complex number|complex]] [[vector space]] {{math|''V''}}, then the '''complex conjugate representation''' {{math|{{overline|Π}}}} is defined over the [[complex conjugate vector space]] {{math|{{overline|V}}}} as follows:
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


:{{math|{{overline|Π}}(''g'')}} is the [[conjugate linear map|conjugate]] of {{math|Π(''g'')}} for all {{math|''g''}} in {{math|''G''}}.
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


{{math|{{overline|Π}}}} is also a representation, as you may check explicitly.
Registered users will be able to choose between the following three rendering modes:


If {{math|'''g'''}} is a [[real number|real]] [[Lie algebra]] and {{math|π}} is a representation of it over the vector space {{math|''V''}}, then the conjugate representation {{math|{{overline|π}}}} is defined over the conjugate vector space {{math|{{overline|''V''}}}} as follows:
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


:{{math|{{overline|π}}(''X'')}} is the conjugate of {{math|π(''X'')}} for all {{math|''X''}} in {{math|'''g'''}}.<ref>This is the mathematicians' convention. Physicists use a different convention where the [[Lie bracket of vector fields|Lie bracket]] of two real vectors is an imaginary vector. In the physicist's convention, insert a minus in the definition.</ref>
<!--'''PNG''' (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


{{math|{{overline|π}}}} is also a representation, as you may check explicitly.
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


If two real Lie algebras have the same [[complexification]], and we have a complex representation of the complexified Lie algebra, their conjugate representations are still going to be different. See [[spinor]] for some examples associated with spinor representations of the [[spin group]]s {{math|Spin(''p'' + ''q'')}} and {{math|Spin(''p'', ''q'')}}.
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


If <math>\mathfrak{g}</math> is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket),
==Demos==


:{{math|{{overline|π}}(''X'')}} is the conjugate of {{math|&minus;π({{overline|''X''}})}} for all {{math|''X''}} in {{math|'''g'''}}
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


For a [[unitary representation]], the dual representation and the conjugate representation coincide.


==See also==
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


*[[Dual representation]]
==Test pages ==


==Notes==
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
<references/>
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


[[Category:Representation theory of groups]]
*[[Inputtypes|Inputtypes (private Wikis only)]]
*[[Url2Image|Url2Image (private Wikis only)]]
==Bug reporting==
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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