MATH 465 (Fall 2010)
## 2010 - 11 Fall

# MATH 465: Geometric Algebra

**Catalogue Description:** General linear groups.
Projective geometry and projective linear groups. Bilinear
forms. Symplectic and
orthogonal geometries. Symplectic groups. Orthogonal groups. Hermitian
forms and unitary groups.
**Prerequisite Courses:** Math 262 and Math 367

**Textbook:** Larry C. Grove, *Classical Groups and Geometric
Algebra*, AMS, 2001.
**Other Books:** Robert A. Wilson, *The Finite Simple Groups*,
Springer, 2009. (Chapter 3)

Steven Roman, *Advanced Linear Algebra*, Springer, 2005. (Chapter
11)

Donald E. Taylor, *The Geometry of Classical Groups*, Heldermann,
1992.

Emil Artin, *Geometric Algebra*, Intersicence, 1957.

**Grading:** Two sets of homework (2 x 30 = 60 points), and one
final exam (40 points). (The final exam will be open-book.)

**Done in the class:**

Week 1: (2 hrs) A quick review of basic algebraic structures. General
linear group. Affine geometry, affine general linear group.

Week 2: Multiple transitivity. Projective lines. GL(2,F) acts
2-transitively on the projective line over F.

Week 3: (1 hr) PGL(2,F) acts sharply 3-transitively on the projective
line over F.

Week 4: Orders of PGL(n), PSL(n) over finite fields. Projective
planes. Number of points and lines in a projective plane over a finite
field. Fano plane. Introduction to bilinear forms. Radicals.

Week 5: Types of bilinear forms, reflexive bilinear forms are either
symmetric or alternate. Structure theorem for alternate forms.

Week 6: Symplectic groups, transvections. Primitve actions.

Week 7: Sp(V) acts primitively on the projective space P(V). Order of
Sp(2m,q).

Week 8: Representation of symmetric bilinear forms, special case over
various fields, quadratic forms.

Week 9: Hyperbolic planes, orthogonal sums of hyperbolic planes.
Orthogonal group, special orthogonal
group, Witt's Cancellation Theorem.

Week 10: Witt's Extension Theorem, maximal totally istropic spaces,
Witt index, structure theorem for quadratic spaces. Description of
O(V), where V is a hyperbolic plane.

Week 11: Z(O(V))={1,-1}. Projective cones. Action of O(V) on its
projective cone.

Week 12: (2 hrs) Hermitian forms, unitary spaces.

Week 13: Unitary group, special unitary group, transvections, their action
on the projective cone.

Week 14: More on SU(V) and PSU(V). Applications in physics by the guest
lecturer
Yusuf �peko�lu.

*Last updated on 5 January 2011.*