Compact symplectic groups and the classification of simple Lie groups
(Disclaimer: This series assumes some familiarity with linear algebra and Lie groups.)
In the previous part, we showed the relation between unit quaternions and three-dimensional rotations, and justified the advantages of this representation for the practical applications. In this part, we will introduce an inner product between vectors of quaternions, and define the compact symplectic groups, the symmetry groups of that inner product. Finally, we will discuss how these groups are fundamental for the classification of simple Lie groups.
The Euclidean inner product on
where
The generalization of this product to
where
These ideas can be naturally generalized to
With these definitions, we are in position to state the classification of simple Lie groups, which are the building blocks of all Lie groups.
exhaust all simple Lie groups. These families are called the classical Lie groups. This classification is a remarkable result of Lie theory. From it, we can see that the compact symplectic groups have as fundamental a role as the unitary and orthogonal groups, another testament to the importance of quaternions.
In the next and final part, we will introduce the automorphism group of quaternions, along with those of the other normed division algebras. This will shed some more light on the relationship of quaternions and three-dimensional rotations.