Lemma. Closed local carriers and continuous Clifford actions [fcap-0045]

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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.