Lie algebras Lie groups Vector spaceapantano/pdf_documents/... · Finite groups Compact groups Lie...

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1 Lie algebras Finite groups Compact groups Lie groups Vector space

Transcript of Lie algebras Lie groups Vector spaceapantano/pdf_documents/... · Finite groups Compact groups Lie...

Page 1: Lie algebras Lie groups Vector spaceapantano/pdf_documents/... · Finite groups Compact groups Lie groups Vector space. 2 Alessandra Pantano Cornell University Finite subgroups of

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Lie algebras

Finite groups

Compact groups

Lie groupsVectorspace

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Alessandra PantanoCornell University

Finite subgroups of SO(3)

Program for Women in Mathematics 2005Institute for Advanced Study, Princeton, NJ

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How do we find finite subgroups?

Rotational symmetriesof regular polyhedra

SO(3)= Rotations in 3D

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Regular Polyhedra

Tetrahedron Cube Octahedron

Dodecahedron Icosahedron

We obtain 3 distinct groups of rotations!!!

S4 S4A4

A5

A5

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Why do we only get 3 groups?

Duality of regular polyhedra: #V #F

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Are there other finite subgroups?

rotational symmetries of other solids…

C n

n-gonalpyramid

D2n

n-gonalplate

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Finite subgroups of SO(3)

D2n

C n

A 5

A 4

S 4

whatelse??

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A beautiful theorem!!!!!

There are no other

finite subgroups

of SO(3)

Theorem: A finite subgroup of SO(3) is either cyclic, or dihedral, or it is the group of rotations of a platonic solid.

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tetrahed. pyramid

# P order stabilizers name realization2 2

lGln n n n C

# O

plate2n+2 3 n 2 n n D cube26 3 24 3 4 4S

14 3 12 2 4 A dodecah.62 3 60 3 2 5 A

2 2

3 3 5

Finite subgroups of SO(3)

G < SO(3) finite

G acts on “poles”

tetrah.

POLE 1

POLE 2

n-fold

axis

2πn

2 1− 1

G⎛⎝⎜

⎞⎠⎟

= 1− 1st Oi( )

⎝⎜

⎠⎟

i=1

#O

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Symmetries of the pyramid

n = # sides in base

# poles = 2

# orbits = 2

1 n-fold axis

G = cyclic group Cn

order of each stabilizer = n

order of the group = n

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Symmetries of the plate

n = # sides

# poles = 2n+2

# orbits = 3

n 2-fold axes

G = dihedral D2n

order of stabilizers= 2, 2, n

order of the group = 2n

1 n-fold axis+

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Symmetries of the cube (8 vertices, 12 edges, 6 faces)

# poles = 26; # orbits = 3

order of stabilizers= 4, 2, 3order of the group = 24

3 4-fold axes

4 3-fold axes

4 2-fold axes

lGl=1+ 3x3+ 4+ 4x2 = 24 = 4!

G permutes the 4 diag.s

G < S 4

lGl = 4! 4G = S

2π4

2π2

2π3

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Symmetries of the tetrahedron(4 vertices, 6 edges, 4 faces)

# orbits = 3order of stabilizers= 2, 3, 3order of the group = 1+ 3 x 1 + 4 x 2 = 12

# poles = 14 (6 centers of an edge, 4 vertices, 4 centers of a face)

G permutes the 4 vertices G < S4

lGl = 4!/2G < A4G generated by (...) and (..)(..)

3 2-fold axes 4 3-fold axes

2π2

2π3

G = A4

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# poles = 62; # orbits = 3

order of stabilizers= 2, 3, 5order of the group = 60

15 2-fold axes

6 5-fold axes

10 3-fold axes

lGl =1+15+10x2 ++ 6x4 = 60 = 5!/2

Symmetries of the dodecahedron(20 vertices, 30 edges, 12 faces)

G = A5

2π2

2π3

2π5

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G permutes the 5 cubes G < S5

G = A5lGl = 60 = 5!/2

G contains all (...) G > A5

Symmetries of the dodecahedron

There areexactly 5 cubes s.t.each edge of the cubeis a diagonalof exactly 1 pentagon.