Continuity of the Spin projection

For a finite-dimensional quadratic space over a topological field with 2 invertible, the Spin projections to the orthogonal and special orthogonal groups are continuous. The groups inherit the canonical topologies from the Clifford algebra and the linear automorphism group, respectively. The latter topology records both the forward and inverse endomorphisms.

The results apply without choosing coordinates or assuming nondegeneracy, separation, or local compactness. They provide the continuous point-group maps used to compare local Spin and special orthogonal subgroups. The inclusion into the Lipschitz group is continuous because a Spin element's inverse in the Clifford algebra is its star; the projections then restrict the existing continuous Lipschitz representation.

References

• H. B. Lawson and M.-L. Michelsohn, Spin Geometry (1989), Chapter I §2.

theorem TauCeti.CliffordAlgebra.continuous_spinToOrthogonal {K : Type u_1} {V : Type u_2} [Field K] [TopologicalSpace K] [IsTopologicalSemiring K] [Invertible 2] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (Q : QuadraticForm K V) : Continuous ⇑(CliffordAlgebra.spinToOrthogonal Q)

The Spin representation into the orthogonal group is continuous for the canonical Clifford-algebra and linear-automorphism topologies.

theorem TauCeti.CliffordAlgebra.continuous_spinToSpecialOrthogonal {K : Type u_1} {V : Type u_2} [Field K] [TopologicalSpace K] [IsTopologicalSemiring K] [Invertible 2] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (Q : QuadraticForm K V) : Continuous ⇑(CliffordAlgebra.spinToSpecialOrthogonal Q)

The Spin projection into the special orthogonal group is continuous for its subgroup topology in the linear automorphism group.