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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
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
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
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
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
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
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
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.
  • 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 e 8
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.”
  • 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.