Continuity of the Pin representation #
For a finite-dimensional quadratic space over a topological field with 2 invertible, the
Pin representation into the orthogonal group is continuous. The domain has the subtype topology
from the Clifford algebra, and the codomain has the subgroup topology from the linear
automorphism group. The result restricts the continuous Lipschitz representation along the
continuous Pin inclusion and enables the later restriction to Spin.
References #
- H. B. Lawson and M.-L. Michelsohn, Spin Geometry (1989), Chapter I §2.
theorem
TauCeti.CliffordAlgebra.continuous_pinToOrthogonal
{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)
:
The Pin representation into the orthogonal group is continuous for the canonical Clifford-algebra and linear-automorphism topologies.