Скачать или смотреть The J function, sl(2) and the Jacobi identity | Universal Hyperbolic Geometry 19 | NJ Wildberger

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. Thanks to EmptySpaceEnterprise for help with contents.

Video Content
00:00 Introduction, the J function, sl(2), the Jacobi identity
05:41 sl(2) Lie algebra
10:07 Jacobi identity
13:52 Projective algebra of matrices
20:20 Projective parametrization of null circle
24:38 General formula for reflection
34:42 The meaning of the Jacobi identity

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