Simple Groups of Lie Type

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# Simple Groups of Lie Type - PowerPoint PPT Presentation

David Renardy. Simple Groups of Lie Type. Simple Groups. Simple Group- A nontrivial group whose only normal subgroups are itself and the trivial subgroup. Simple groups are thought to be classified as either: Cyclic groups of prime order (Ex. G=<p>)

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## Simple Groups of Lie Type

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Presentation Transcript
David Renardy

### Simple Groups of Lie Type

Simple Groups
• Simple Group- A nontrivial group whose only normal subgroups are itself and the trivial subgroup.
• Simple groups are thought to be classified as either:
• 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.
Lie Groups
• 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.
Lie Groups (cont’d)
• 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)
Simple Lie Groups
• Definition: A connected lie group that is also simple.
• Connected: Topological concept, cannot be broken into disjoint nonempty closed sets.
• 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 Lie-Type 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
Classification of Simple Lie Groups
• 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)
Exceptional Cases
• 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
Simple Groups of Lie Type
• Classical Groups
• Special Linear, orthogonal, symplectic, or unitary group.
• Chevalley Groups
• Defined Simple Groups of Lie Type over the integers by constructing a Chevalley basis.
• Steinberg Groups
• Completed the classical groups with unitary groups and split orthogonal groups
• the unitary groups2An, from the order 2 automorphism of An;
• further orthogonal groups2Dn, 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.
E8
• 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)
Science and E8
• 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.”
Sources
• http://cache.eb.com/eb/image?id=2106&rendTypeId=4
• http://aimath.org/E8/images/e8plane2a.jpg
• http://www.mpa-garching.mpg.de/galform/press/seqD_063a_small.jpg
• http://superstruny.aspweb.cz/images/fyzika/aether/honeycomb.gif
• Wikipedia.org
• Mathworld.com
• Aschbacher, Michael. The Finite Simple Groups and Their Classification. United States: Yale University, 1980.