Quaternions and Lie groups (part 3)

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 $\mathbb{R}^n$ is

\[\langle \mathbf{x},\mathbf{y} \rangle = \sum_i x_i y_i\,,\]

where $\mathbf{x} = (x_1, \ldots, x_n)$ and $\mathbf{y} = (y_1, \ldots, y_n)$ are vectors in $\mathbb{R}^n$. The orthogonal group, $\mathrm{O}(n)$, can be regarded as the group of real matrices which preserve the Euclidean inner product.

The generalization of this product to $\mathbb{C}^n$ is the Hermitian inner product,

\begin{equation} \langle \mathbf{z},\mathbf{w} \rangle = \sum_i z_i \overline{w_i}\,, \label{hermitian-inner-product} \end{equation}

where $\overline{w}$ is the complex conjugate of $w$. In this case, the unitary group, $\mathrm{U}(n)$, can be regarded as the group of matrices in $\mathrm{M}_n (\mathbb{C})$ which preserve the Hermitian product.

These ideas can be naturally generalized to $\mathbb{H}^n$, simply by interpreting $\overline{w}$ in the Hermitian product above as a quaternion conjugate, and $z$ and $w$ as vectors in $\mathbb{H}^n$. This defines a standard inner product on $\mathbb{H}^n$, named the symplectic inner product. Thus, we define the compact symplectic group, $\mathrm{Sp}(n)$, as the group of matrices in $\mathrm{M}_n (\mathbb{H})$ which preserve the symplectic inner product on $\mathbb{H}^n$. It’s not difficult to see that $\mathrm{Sp}(n)$ is a matrix Lie group.

With these definitions, we are in position to state the classification of simple Lie groups, which are the building blocks of all Lie groups.The precise meaning of this requires some concepts that I only define here for reference. A connected Lie group is called simple if its Lie algebra is simple, i.e. it's non-abelian and contains no nonzero proper ideals. This is equivalent to the Lie group being non-abelian and having no nontrivial connected normal subgroups (note that it can have discrete normal subgroups). A Lie algebra is called semisimple if it's a direct sum of simple Lie algebras. On the other hand, the derived algebra of a Lie algebra $\mathfrak{g}$ is the subalgebra $[\mathfrak{g},\mathfrak{g}]$, i.e. the subalgebra of commutators of vectors in $\mathfrak{g}$. The derived series of $\mathfrak{g}$ is the series of subalgebras $\mathfrak{g} \geq [\mathfrak{g},\mathfrak{g}] \geq [[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]] \geq ...$ where each subalgebra is the derived algebra of the previous one. A Lie algebra is called solvable if its derived series ends in the zero algebra. In particular, this implies that its derived algebra is nilpotent. The Levi decomposition theorem states that any Lie algebra can be decomposed as the semidirect product of a semisiple Lie algebra and a solvable Lie algebra. Via the Lie correspondence, this means any simply connected Lie group is decomposed as the semidirect product of a semisimple Lie group and a solvable Lie group. Finally, all connected Lie groups are obtained as quotients of the simply connected Lie groups by their discrete normal subgroups. Apart from five exceptional Lie groups, the four infinite families

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.