### UnivHypGeom19: The J function, sl(2) and the Jacobi identity

We review the basic connection between hyperbolic points and matrices, and connect the J function, which computes the joins of points or the meets of lines, with the Lie bracket of 2x2 matrices. This connects with the Lie algebra called sl(2) in the projective setting. The Jacobi identity then gives a new proof of the concurrence of the altitudes of a triangle, in other words the existence of the orthocenter.CONTENT SUMMARY: pg 1: @00:11 Introduction; the J? function, sl(2), the Jacobi identitypg 2: @05:41 sl(2) Lie algebra in a nutshellpg 3: @10:07 Jacobi Identity; proof; simpler identitypg 4: @13:52 Projective algebra of matrices;pg 5: @20:20 Review of connection between matrices and points and lines; Projective parametrization of null circle; Important - hyperbolic points are associated to projective trace zero matricespg 6: @24:38 Continued review; General formula for reflection; Bracket? theorem; the bracket computes the J functionpg 7: @28:46 proof of Bracket theorempg 8: @34:42 The meaning of the Jacobi identity (THANKS to EmptySpaceEnterprise)

Length:
42:27