Simple Groups of Lie Type

1. David Renardy Simple Groups of Lie Type

2. 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.

3. 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.

4. 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)

5. 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

6. 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)

7. 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

8. 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.

9. 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)

10. 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.”

11. 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.