Geometry & Topology 2024, 34: Quaternions, Rotations and the Geometry of Transformation Groups

A topological group is a group on which multiplication and inverse are continuous. The openness of topology does most of the work, as from that, and the homeomorphism properties, the topology follows from openness/compactness. We take the orthogonal group O(n) as a compact topological space axiomatically by proofs that happened earlier, and due to its common function as a symmetry gauge group. We remember that elements of GL(n) can be written as products of an orthogonal matrix and an upper triangle matrix with positive diagonal entries. The map decomposing the GL(n) matrix is a homeomorphism.

The algebra of quaternions is the real vector space

where ijk are indices in the Levi-Civita symbol. H is an associative, noncommutative real algebra with 4 dimensions, where conjugation is an anti-involution: (pq)* = q*p* and the inverse is 2-sided. It's a division algebra/skew field. Define the unit Quaternions as

The maps a, b are direct motions of H, fixing the origin, and r(x) = qxq* is a rotation on E³. Any element q of U which is not real, has a unique decomposition q = cos(θ) + I sin(θ) where I is a pure imaginary quaternion, and θ ∈ (0, π). There is a homeomorphism between SO(3) and S³/~ and between SO(4) and (S³ ⨉ S³)/≈. The group of unit quaternions can be written as a matrix group and isomorphically mapped onto SU(2), which is the symmetry group formally associated with spin