Split-Complex Numbers as Clifford Algebra, Direct Sum and Matrix Algebra

Split-complex numbers have the form x + y j where x and y are real numbers and j^2 = 1. They are an important example of a Clifford algebra. They are isomorphic to the direct sum R+R. I present this isomorphism. I show various ways to express them as a matrix algebra. In particular, I calculate the simple R+R-modules. Here are my notes https://www.math4wisdom.com/wiki/Research/SplitComplexNumbers Understanding the split-complex numbers is one step in understanding Bott periodicity. https://www.math4wisdom.com/wiki/Exposition/20231126BottPeriodicity This is part of my exploration of how to mathematically model the ways that our minds carve up divisions of everything. Recorded 2024.04.27. 0:00 Introduction 4:00 Overview of notes 7:45 Extending the real numbers. Algebraic structures: Group, vector space, ring, algebra. 22:30 Comparing split-complex numbers with complex numbers and dual numbers. 26:50 Comparing geometry of these numbers. 31:20 Motivation: Connection with Bott periodicity, how mental perspectives are related. 39:10 Summarizing earlier video on Bott periodicity, Clifford algebras, CPT symmetry and modeling consciousness. 49:50 Morita equivalence. More about complex and dual numbers. 52:50 Expressing a Clifford algebra as a matrix algebra. 59:20 Direct sum: R+R. 1:03:00 Determining the two simple R+R-modules (a,b)(c) = ac and (a,b)(c)=bc. Thank you to John Baez at the Category Theory ZulipChat! 1:06:40 Show that they satisfy the rules for modules. 1:08:30 Show that every element of C_1 has the form rc_1. 1:16:15 Show these are simple modules. 1:18:30 How do we know these two modules are not the same? 1:22:10 Show there is no other simple module, up to isomorphism. 1:24:50 An isomorphism between the Clifford algebra and the direct sum R+R. 1:31:40 Comparing 2x2 matrix algebra representations for split-complex numbers, complex numbers and dual numbers. Determinants. 1:42:15 Calculating eigenvalues and eigenvectors. 1:47:35 Change of basis. Diagonalizability. 1:49:45 Understanding the direct sum in terms of 2x2 matrices. The two projections as 2x2 matrices. 1:54:45 The two projections as 1x1 matrices. 1:57:20 Summary and upcoming research! Join our Math 4 Wisdom email group https://www.freelists.org/list/math4wisdom to participate in our community and learn how to join our Math 4 Wisdom study groups. Thank you for leaving comments, liking this video and subscribing to this M4W YouTube channel! Thank you, Daniel Friedman, John Harland, Bill Pahl and all for your support through Patreon! https://www.patreon.com/math4wisdom Visit http://www.math4wisdom.com to contact me, Andrius Kulikauskas, and learn more about Math4Wisdom!