Lie Groups and Lie Algebras: Lesson 2 - Quaternions

This video is about Lie Groups and Lie Algebras: Lesson 2 - Quaternions

We study the algebraic nature of quaternions and cover the ideas of an algebra and a field. Later we will discover how quaternions fit into the description of the classical Lie Groups.

NOTE: An astute viewer noted that I identified the quaternions as an algebraic field despite the fact that quaternions are not commutative! Gilmore, the author of the book we are following defines fields a bit more generally than most authors. Gilmore asserts that, basically, that a non commutative division ring is a "field" and if the elements of the division ring do commute then the ring is a "commutative field." Almost every modern reference however includes commutativity as a field axiom. Also, in my previous lecture on the subject (What is a Tensor lesson 20 I think "Algebraic Structures II") I used the modern definition! Sometimes a field like the quaternions is called a "Skew field".

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