1. David Renardy

2.  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= ) ▪ Alternating groups of degree at least 5. (Ex. A 5 ) ▪ Groups of Lie Type (Ex. E 8 ) ▪ One of the 26 Sporadic groups (Ex. The Monster)  First complete proof in the early 90’s, 2 nd Generation proof running around 5,000 pages.

3.  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.

4.  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)

5.  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

6.  Infinite families  A n series corresponds to the Special Unital Groups SU(n+1) (nxn unitary matrices with unit determinant)  B n series corresponds to the Special Orthogonal Group SO(2n+1) (nxn orthogonal matrices with unit determinatnt)  C n series corresponds to the Symplectic (quaternionic unitary) group Sp(2n)  D n series corresponds to the Special Orthogonal Group SO(2n)

7.  G 2 has rank 2 and dimension 14 G 2  F 4 has rank 4 and dimension 52F 4  E 6 has rank 6 and dimension 78E 6  E 7, has rank 7 and dimension 133E 7  E 8, has rank 8 and dimension 248E 8

8.  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 groups 2 A n, from the order 2 automorphism of A n ; ▪ further orthogonal groups 2 D n, from the order 2 automorphism of D n ; ▪ the new series 2 E 6, from the order 2 automorphism of E 6 ; ▪ the new series 3 D 4, from the order 3 automorphism of D 4.

9.  We can represent groups of Lie type by their “root system” or a set of vectors spanning R n where n is the rank of the Lie algebra, that satisfy certain geometric constraints.  The E 8 group can be represented in an “even coordinate system” of R 8 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)

10.  Applications in Theoretical Physics relate to String Theory and “supergravity”  “The group E 8 ×E 8 (the Cartesian product of two copies of E 8 ) 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.”Cartesian productgauge group heterotic stringanomaly-freesupergravity

11.  http://cache.eb.com/eb/image?id=2106&rendTypeId= 4 http://cache.eb.com/eb/image?id=2106&rendTypeId= 4  http://aimath.org/E8/images/e8plane2a.jpg http://aimath.org/E8/images/e8plane2a.jpg  http://www.mpa- garching.mpg.de/galform/press/seqD_063a_small.jpg http://www.mpa- garching.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.