Lemma. Closed local carriers and continuous Clifford actions [fcap-0045]
AGENTDRAFTED
Definition [spin-000O]Let \(K\) be a local field of characteristic different from two in the
sense of Definition [fcap-0044], and let \(Q\) be a nondegenerate quadratic form on
a finite-dimensional \(K\)-space \(V\). Give all spaces and carriers the
canonical topologies of that definition. Then \(O(Q)\) and \(SO(Q)\) are
closed in \(\operatorname {End}_K(V)\), and the actual vector-generated
\(\operatorname {Spin}(Q)\) is closed in \(A=\mathcal {C}\kern -2pt\ell (Q)\). These are locally compact Hausdorff
topological groups. Their actions on \(V\) and the projection
\(\rho :\operatorname {Spin}(Q)\to SO(Q)\) are continuous.
Assume first that \(n=\dim _K V>0\). Write \(\widehat v=\iota _Q(v)\),
\(\alpha \) for the grading involution, and \(\operatorname {rev}\) for
reversion. The vector-generated
Lipschitz group has the following closed description
inside \(A^\times \):
\[\begin {gathered}
\Gamma (Q)=\{x\in A^\times :\tau _x(V)\subseteq V\},\\
\tau _x(\widehat v)=\alpha (x)\widehat v x^{-1}.
\end {gathered}\]
Here and below \(V\) is identified with its embedded vector subspace. We
verify this description so that the generated carrier is retained. If
\(w=\alpha (x)\widehat v x^{-1}\in V\), applying \(\alpha \) also gives
\(w=x\widehat v\alpha (x)^{-1}\). Multiplying the two expressions yields
\(w^2=Q(v)1\). Thus \(\tau _x\) preserves \(Q\); nondegeneracy makes it
injective, hence an isometry of \(V\). By Proposition [fcap-002D], choose a vector
product \(y\in \Gamma (Q)\) with this action. For \(z=y^{-1}x\),
\[\alpha (z)\widehat v=\widehat v z\qquad (v\in V).\]
In an orthogonal basis \(e_1,\ldots ,e_n\), expand \(z\) in the Clifford
monomials \(e_I\). For a fixed \(j\), the displayed equation is automatic
on monomials with \(j\notin I\); on those with \(j\in I\), its two sides
have opposite signs. The nonzero factors \(2Q(e_j)\) and linear
independence force all their coefficients to vanish. Doing this for
every \(j\) leaves only a scalar \(z=c1\), with \(c\ne 0\). For any
\(Q(v)\ne 0\),
\[c1=\widehat v\,\widehat {\left (\frac {c}{Q(v)}v\right )}.\]
So \(c1\), and hence \(x\), belongs to \(\Gamma (Q)\). The reverse inclusion
holds because each vector unit acts as a reflection. This also explains
the general-field carrier argument behind the real case in Proposition [fcap-0037].
The actual Spin carrier is therefore
\[\begin {gathered}
\operatorname {Spin}(Q)=\{x\in A^{\mathrm {even}}:\\
\operatorname {rev}(x)x=1,\\
x\widehat v\operatorname {rev}(x)\in V\quad (v\in V)\}.
\end {gathered}\]
In finite dimension this left inverse is two-sided:
\(L_{\operatorname {rev}(x)}L_x=1\) makes \(L_x\) invertible and gives
\(x\operatorname {rev}(x)=1\). The preceding description now puts \(x\)
in \(\Gamma (Q)\); evenness and reverse norm one are exactly the Spin
conditions of Remark [fcap-0028]. Conversely every Spin element satisfies
these equations. This is the norm-one convention: the star and reverse
norms agree on even elements.
Every finite-dimensional linear subspace of \(A\) is closed: extend
a basis and set the complementary coordinates equal to zero. Evenness
and membership in \(V\) are consequently closed conditions. Reversion is
linear, and multiplication is polynomial. It suffices to impose vector
preservation on a basis of \(V\), so the displayed Spin carrier is a
finite intersection of closed polynomial conditions in \(A\). The same
argument, with continuous inversion on \(A^\times \), proves that
\(\Gamma (Q)\) is closed in \(A^\times \).
Choose a basis of \(V\) and let \(G\) be the nonsingular Gram matrix
of the half-polar form \(b_Q\) from Convention [fcap-0027]. Then
\[\begin {aligned}
O(Q)&=\{g:g^{\mathsf T}Gg=G\},\\
SO(Q)&=\{g\in O(Q):\det g=1\}.
\end {aligned}\]
The first equation already forces \(g\) invertible by taking determinants.
These polynomial equations prove closedness in the whole matrix space.
Matrix inversion, Clifford inversion and multiplication are continuous
by Definition [fcap-0044]; their restrictions give the topological group
operations. Finite-dimensional spaces over \(K\) are locally compact
Hausdorff, so the closed carriers have those properties.
The joint action \((x,v)\mapsto \alpha (x)\widehat v x^{-1}\)
is continuous by the same coordinate operations. Its values lie in the
closed vector subspace, whose inverse identification with \(V\) is linear
and continuous. Evaluating on a basis proves continuity of
\(\tau :\Gamma (Q)\to O(Q)\) in matrix coordinates. Restricting to Spin
removes \(\alpha \), and even determinant parity from Lemma [fcap-001A]
places its image in \(SO(Q)\), proving continuity of \(\rho \). The joint
matrix actions of \(O(Q)\) and \(SO(Q)\) are polynomial as well.
If \(V=0\), its only vector has zero quadratic value. There are no
invertible vector generators, so \(\Gamma (Q)=\operatorname {Spin}(Q)=\{1\}\); both
orthogonal groups are also trivial. All conclusions follow for these
one-point carriers. The positive-dimensional normalizer description must
not be used here: every Clifford unit preserves the zero vector space.
TauCeti proves closedness and local compactness for the actual carrier
in CliffordAlgebra.isClosed_spinGroup and
CliffordAlgebra.locallyCompactSpace_spinGroup.
The continuous projection is
TauCeti.CliffordAlgebra.continuous_spinToSpecialOrthogonal.
These declarations use canonical module and subgroup topologies; the first
two allow Hausdorff topological fields, with local compactness added for
the second. The projection itself needs neither nondegeneracy nor local
compactness. The argument above supplies their coordinate specialization.