Wat is Discrete Algebra & Meetkunde? - Magma
Wat is Discrete Algebra & Meetkunde? - Magma
Wat is Discrete Algebra & Meetkunde? - Magma
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Algorithms<br />
for Lie <strong>Algebra</strong>s<br />
of <strong>Algebra</strong>ic Groups
<strong>Wat</strong> <strong>is</strong> D<strong>is</strong>crete <strong>Algebra</strong> & <strong>Meetkunde</strong>?<br />
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<strong>Wat</strong> <strong>is</strong> D<strong>is</strong>crete <strong>Algebra</strong> & <strong>Meetkunde</strong>?<br />
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<strong>Wat</strong> <strong>is</strong> D<strong>is</strong>crete <strong>Algebra</strong> & <strong>Meetkunde</strong>?<br />
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<strong>Wat</strong> <strong>is</strong> D<strong>is</strong>crete <strong>Algebra</strong> & <strong>Meetkunde</strong>?<br />
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<strong>Wat</strong> <strong>is</strong> D<strong>is</strong>crete <strong>Algebra</strong> & <strong>Meetkunde</strong>?<br />
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<strong>Wat</strong> <strong>is</strong> D<strong>is</strong>crete <strong>Algebra</strong> & <strong>Meetkunde</strong>?<br />
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<strong>Wat</strong> <strong>is</strong> D<strong>is</strong>crete <strong>Algebra</strong> & <strong>Meetkunde</strong>?<br />
•<br />
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Drie problemen, die eigenlijk hetzelfde zijn<br />
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•De symmetrie wordt beschreven door een groep:<br />
de dihedrale groep van orde 4<br />
•<br />
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Uitspraken over de groep<br />
⇒ uitspraken over<br />
alledrie de problemen
Een heleboel structuren<br />
groep<br />
algebra<br />
Lie algebra<br />
algebraïsche groep<br />
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lichaam<br />
ring<br />
polynoomring<br />
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eindig lichaam<br />
moduul<br />
monoïde<br />
wortelsysteem<br />
D<strong>is</strong>crete w<strong>is</strong>kunde bestudeert dit soort<br />
structuren en hun onderlinge relaties
Lie <strong>Algebra</strong>s<br />
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L<br />
2<br />
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Lie <strong>Algebra</strong>s<br />
lichaam<br />
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2<br />
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vermenigvuldiging<br />
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Lie <strong>Algebra</strong>s<br />
lichaam<br />
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2<br />
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vermenigvuldiging<br />
0111011011111000001 0010000011101110110 0000000000000000000 0000000000000000000 0000000000000000000
Lie <strong>Algebra</strong>s<br />
0000001000011000000<br />
L<br />
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Lie <strong>Algebra</strong>s<br />
L<br />
•Enkelvoudige (simpele) Lie algebras ...<br />
•... zijn geclassificeerd.<br />
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An<br />
An An<br />
An An<br />
An An<br />
An<br />
An<br />
An Bn<br />
An Cn<br />
An Dn<br />
An<br />
2<br />
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E6 E7 E8 F4 G2
Lie <strong>Algebra</strong>s<br />
L<br />
•Enkelvoudige (simpele) Lie algebras ...<br />
•... zijn geclassificeerd.<br />
0000001000011000000<br />
0000001000011000000<br />
An<br />
An An<br />
An An<br />
An An<br />
An<br />
An<br />
An Bn<br />
An Cn<br />
An Dn<br />
An<br />
22<br />
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E6 E7 E8 F4 G2
Herkennen van Lie <strong>Algebra</strong>s<br />
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•Welke Lie algebra <strong>is</strong> dit eigenlijk?<br />
•An, Bn, Cn, Dn, E6, E7, E8, F4, G2 ??
Herkennen van Lie <strong>Algebra</strong>s<br />
In veel gevallen:<br />
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?? G2<br />
!<br />
2<br />
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Herkennen van Lie <strong>Algebra</strong>s<br />
In veel gevallen:<br />
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?? G2 !<br />
Maar soms:<br />
2<br />
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?????
Mijn onderzoek<br />
In heel veel gevallen,<br />
ook over “slechte” lichamen:<br />
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Algoritmen voor het vinden van gespleten torale<br />
F4<br />
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deelalgebras en Chevalley bases, met toepassing op<br />
herkenning van Lie algebras en het bewijs dat een<br />
bepaalde graaf niet afstandstransitief <strong>is</strong>.<br />
!
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