The J Function, SL(2) and the Jacobi Identity - Universal Hyperbolic Geometry

Explore the connections between hyperbolic geometry, matrix algebra, and Lie theory in this advanced mathematics lecture. Delve into the J function, which computes joins of points and meets of lines, and its relationship to the Lie bracket of 2x2 matrices. Examine the sl(2) Lie algebra in a projective setting and discover how the Jacobi identity provides a novel proof for the concurrence of triangle altitudes. Investigate projective matrix algebra, the parametrization of null circles, and general reflection formulas. Gain insights into the deeper meaning of the Jacobi identity and its implications for hyperbolic geometry.

Introduction, the J function, sl2, the Jacobi identity
sl2 Lie algebra
Jacobi identity
Projective algebra of matrices
Projective parametrization of null circle
General formula for reflection
The meaning of the Jacobi identity