Transcript Simple Groups of Lie Type

David Renardy
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Simple Group- A nontrivial group whose only
normal subgroups are itself and the trivial
subgroup.
 Simple groups are thought to be classified as either:
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Cyclic groups of prime order (Ex. G=<p>)
Alternating groups of degree at least 5. (Ex. A5 )
Groups of Lie Type (Ex. E8 )
One of the 26 Sporadic groups (Ex. The Monster)
 First complete proof in the early 90’s, 2nd Generation
proof running around 5,000 pages.
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Named after Sophus Lie (1842-1899)
Definition: A group which is a
differentiable manifold and whose
operations are differentiable.
 Manifold- A mathematical space where
every point has a neighborhood
representing Euclidean space. These
neighborhoods can be considered
“maps” and the representation of the
entire manifold, an “atlas” (Ex. Using
maps when the earth is a sphere)
 Differentiable Manifolds-Manifolds
where transformations between maps
are all differentiable.
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Examples:
 Points on the Real line under addition
 A circle with arbitrary identity point and
multiplication by Θ mod2π representing the
rotation of the circle by Θ radians.
 The Orthogonal group (set of all orthogonal nxn
matrices.)
 Standard Model in particle physics
U(1)×SU(2)×SU(3)
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Definition: A connected lie group that is also simple.
 Connected: Topological concept, cannot be broken into disjoint
nonempty closed sets.
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Lie-Type Groups- Many Lie groups can be defined as
subgroups of a matrix group. The analogous subgroups
where the matrices are taken over a finite field are called LieType Groups.
Lie Algebra- Algebraic structure of Lie groups. A vector
space over a field with a binary operation satisfying:
 Bilinearity [ux+vy,w]=u[x,w]+v[y,w]
 Anticommutativity [x,y]=-[y,x]
[x,x]=0
 The Jacobi Identity [x,(y,z)]+[y,(z,x)]+[z,(x,y)]=0
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Infinite families
 An series corresponds to the
Special Unital Groups SU(n+1) (nxn
unitary matrices with unit
determinant)
 Bn series corresponds to the
Special Orthogonal Group
SO(2n+1) (nxn orthogonal matrices
with unit determinatnt)
 Cn series corresponds to the
Symplectic (quaternionic unitary)
group Sp(2n)
 Dn series corresponds to the
Special Orthogonal Group SO(2n)
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G2 has rank 2 and
dimension 14
F4 has rank 4 and
dimension 52
E6 has rank 6 and
dimension 78
E7, has rank 7 and
dimension 133
E8, has rank 8 and
dimension 248
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Classical Groups
 Special Linear, orthogonal, symplectic, or unitary group.
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Chevalley Groups
 Defined Simple Groups of Lie Type over the integers by constructing a
Chevalley basis.
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Steinberg Groups
 Completed the classical groups with unitary groups and split
orthogonal groups
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the unitary groups 2An, from the order 2 automorphism of An;
further orthogonal groups 2Dn, from the order 2 automorphism of Dn;
the new series 2E6, from the order 2 automorphism of E6;
the new series 3D4, from the order 3 automorphism of D4.
We can represent groups of Lie type by
their “root system” or a set of vectors
spanning Rn where n is the rank of the
Lie algebra, that satisfy certain
geometric constraints.
 The E8 group can be represented in an
“even coordinate system” of R8 as all
vectors with length √2 with
coordinates integers or half-integers
and the sum of all coordinates even.
This gives 240 root vectors.
 (±1, ±1,0,0,0,0,0,0) gives 112 root
vectors by permutation of coordinates
(8!/(2!*6!) *4 (for signs))
 (±1/2,±1/2,±1/2,±1/2,±1/2,±1/2,±1/2,
±1/2) gives 128 root vectors by
switching the signs of the coordinates
(2^8/2)
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Applications in Theoretical Physics relate to String
Theory and “supergravity”
 “The group E8×E8 (the Cartesian product of two copies of
E8) serves as the gauge group of one of the two types of
heterotic string and is one of two anomaly-free gauge
groups that can be coupled to the N = 1 supergravity in 10
dimensions.”
http://cache.eb.com/eb/image?id=2106&rendTypeId=
4
 http://aimath.org/E8/images/e8plane2a.jpg
 http://www.mpagarching.mpg.de/galform/press/seqD_063a_small.jpg
 http://superstruny.aspweb.cz/images/fyzika/aether/ho
neycomb.gif
 Wikipedia.org
 Mathworld.com
 Aschbacher, Michael. The Finite Simple Groups and
Their Classification. United States: Yale University,
1980.
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