Example. The split three-dimensional Spin action is symmetric matrix congruence [fcap-003X]
✍️source

For \(Q_{2,1}(v)=v_0^2+v_1^2-v_2^2\), keep the algebra map \(\Phi :\mathcal {C}\kern -2pt\ell ^+(Q_{2,1})\simeq M_2(\mathbb R)\) of Example [fcap-002I]. Multiplication by the last generator gives symmetric matrix coordinates \[X(v)=\Phi (ve_2)= \begin {pmatrix}-v_2-v_1&v_0\\v_0&-v_2+v_1\end {pmatrix}, \qquad \det X(v)=-Q_{2,1}(v).\] The inverse sends \(\begin {pmatrix}a&b\\b&d\end {pmatrix}\) to \((b,(d-a)/2,-(a+d)/2)\). Thus \(X\) is a real linear equivalence onto the symmetric matrices, with the stated negative determinant form TauCeti.coe_realCliffordTwoOneVectorEquivSymmetric_apply TauCeti.realCliffordTwoOneVectorEquivSymmetric_det. For \(s\in \operatorname {Spin}(Q_{2,1})\) and \(M=\Phi (s)\in SL_2(\mathbb R)\), the vector action becomes \[X(svs^{-1})=M X(v)M^{\mathsf T}.\]

To prove the formula, write \(A=e_0e_1\), \(B=e_0e_2\), and \(C=e_1e_2\). Since \(e_2^2=-1\), conjugation by \(e_2\) is \(\gamma (x)=-e_2xe_2\). It fixes \(A\) and negates \(B,C\). On their three matrices in Example [fcap-002I], and on the identity, this gives \[\Phi (\gamma (x))=\operatorname {adj}(\Phi (x))^{\mathsf T}.\] Both sides are linear in \(x\), so the basis check proves the equality on the whole even algebra. Moreover, \[(svs^{-1})e_2=s(ve_2)\gamma (s^{-1}).\] The determinant-one equation gives \(\operatorname {adj}(M^{-1})=M\); applying \(\Phi \) proves the action. TauCeti's bilinear-generator formula sends \(A,B,C\) to exactly these three matrices, so its group map and action use this same \(\Phi \) TauCeti.realCliffordTwoOneEvenEquivMatrix_ι TauCeti.realSpinTwoOneEquivSpecialLinear_action.

For example, let \(M=\operatorname {diag}(a,a^{-1})\) with \(a>0\), and choose its Spin preimage using Example [fcap-002I]. Put \(c=(a^2+a^{-2})/2\) and \(h=(a^2-a^{-2})/2\). The congruence action is \[v_0'=v_0,\qquad v_1'=cv_1+hv_2,\qquad v_2'=hv_1+cv_2.\] Here \(c^2-h^2=1\). The two null coordinates \(v_1+v_2\) and \(v_1-v_2\) are multiplied by \(a^2\) and \(a^{-2}\), respectively. This exhibits the action on both null directions in the chosen coordinates.

For comparison, the compact quaternion action of Example [fcap-003J] has a complex matrix realization. Distinguish quaternion units \(\mathbf i,\mathbf j,\mathbf k\) from the complex scalar \(i\). The map \[\iota (a+b\mathbf i+c\mathbf j+d\mathbf k)= \begin {pmatrix}a+ib&c+id\\-c+id&a-ib\end {pmatrix}\] preserves multiplication and sends quaternion conjugation to conjugate transpose. Its determinant is \(a^2+b^2+c^2+d^2\), so it identifies the unit quaternions with \(SU(2)\). Indeed a special unitary matrix has the form \(\begin {pmatrix}z&w\\-\bar w&\bar z\end {pmatrix}\) with \(|z|^2+|w|^2=1\), which recovers the four real coordinates. These are Gallier's axes [gallier2014clifford, Section 1.1, pp. 8--9], and the exact embedding and group equivalence in TauCeti Quaternion.toComplexMatrix Quaternion.unitaryEquivSpecialUnitaryGroup.

Keep \(\eta _3(v)=-v_1\mathbf i+v_0\mathbf j-v_2\mathbf k\) from Example [fcap-003J], and put \(K(v)=i\iota (\eta _3(v))\). It is a traceless Hermitian matrix. Applying \(i\iota \) to the already proved quaternion action gives \[K(svs^{-1})=U K(v)U^*,\qquad U=\iota (\theta (s))\in SU(2),\] where \(U^*\) is conjugate transpose. The composite Spin-to-\(SU(2)\) isomorphism is available in TauCeti TauCeti.realSpinThreeEquivSpecialUnitary. Its action theorem uses the unscaled matrix \(\iota (\eta _3(v))\) TauCeti.realSpinThreeEquivSpecialUnitary_action; multiplication by the central scalar \(i\) gives the displayed \(K\) formula. Thus neither quaternion nor vector axes change. The split formula uses real symmetric matrices and \(SL_2(\mathbb R)\) instead. The fixed-vector transport underlying these formulas is illustrated by [porteous1995clifford, Chapter 16, Proposition 16.9, pp. 143--144]. For the indefinite vector-action image, retain the spinor-norm kernel of Theorem [fcap-0023]; no component or covering-space assertion is needed.

Example. The split three-dimensional Spin action is symmetric matrix congruence [fcap-003X]
✍️source

Reference. Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions: The Pin and Spin Groups [gallier2014clifford]

Reference. Clifford algebras and the classical groups [porteous1995clifford]

Hyperbolic recurrence, real signatures and quaternary centers [fcap-0010]

Convention. Real signature and Clifford signs [chevalley1954algebraic, II.2.9, pp. 65--66] [fcap-0011]

realCliffordForm realCliffordForm_apply
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Theorem. Witt extension and the regular-subspace interface [fcap-002A]

Proposition. A Lorentz frame can be prescribed [fcap-002B]

Proof.

Remark. The radical blocks an unqualified extension theorem [fcap-002C]

Definition. The hyperbolic plane [lawson2016spin, I.4, Theorem 4.1 and (4.3), pp. 25--26] [fcap-0012]

realCliffordForm_one_one_apply realBottSplitIsometry
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Theorem. The hyperbolic matrix recurrence [lawson2016spin, I.4, Theorem 4.1, (4.3), pp. 25--26] [fcap-0013]

hyperbolicEquivTensor realCliffordBottEquiv_ι
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Lemma. Finite hyperbolic reduction [chevalley1954algebraic, II.2.9, pp. 65--66] [fcap-0014]

realCliffordBottIterEquiv realCliffordSignatureReductionEquiv
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Theorem. The signature-switch recurrence [lawson2016spin, I.4, Theorem 4.1 and (4.1), pp. 25--26] [fcap-001X]

realCliffordSignatureSwitchRecurrenceEquiv
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Remark. Why algebraic periodicity is a separate branch [chevalley1954algebraic, II.2.1, II.2.5, and II.2.9, pp. 42--46, 65--66]; [lawson2016spin, I.4, Theorems 4.1 and 4.3, pp. 25--29] [fcap-001Y]

Example. Four real Clifford base entries [lawson2016spin, I.4, Theorem 4.3 and Tables I--II, pp. 27--29] [fcap-0015]

realCliffordOneZeroEquivProd realCliffordZeroOneEquivComplex realCliffordZeroTwoEquivQuaternion realCliffordOneOneEquivMatrix
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Lemma. Reverse norm one is the Spin carrier in real dimension three [fcap-002G]

Example. The compact three-dimensional Spin group is the unit quaternions [fcap-002H]

Example. The compact three-dimensional Spin action is quaternion conjugation [fcap-003J]

Example. The split three-dimensional Spin group is \(SL_2(\mathbb R)\) [fcap-002I]

Example. The split three-dimensional Spin action is symmetric matrix congruence [fcap-003X]
✍️source

Lemma. Reverse norm one is the Spin carrier in real dimension four [fcap-002J]

Example. The compact four-dimensional Spin group is a pair of unit quaternions [fcap-002K]

Example. The compact four-dimensional Spin action is left and right quaternion multiplication [fcap-003K]

Example. The split four-dimensional Spin action is left and inverse-right multiplication [fcap-003Y]

Example. The Lorentzian four-dimensional Spin group is \(SL_2(\mathbb C)\) [fcap-002L]

Lemma. Opposite Lorentz signatures share an even-algebra action dictionary [fcap-003Z]

Example. The Lorentz Spin action is Hermitian matrix congruence [fcap-0040]

Proposition. The discriminant center of a four-dimensional Spin group [fcap-0043]

Proof.

Lemma. The volume term does not obstruct Spin in dimension five [fcap-003H]

Example. A norm-one even unit can leave the vector space in dimension six [fcap-003I]

Remark. From real recurrences to the residue table [fcap-0016]

Example. The real Clifford algebra of signature (1,3) [fcap-0026]

realClifford_classification_of_residue_eq_two
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Proposition. The real algebra generated by the Dirac gamma matrices [fcap-002U]

Example. The compact three-dimensional Spin action is quaternion conjugation [fcap-003J]

Definition. Spin group [fulton2013representation] [spin-000O]

Theorem. The Spin image is the spinor-norm kernel [fcap-0023]

range_spinToSpecialOrthogonal_eq_ker_spinorNorm
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Proof.

Example. The split three-dimensional Spin group is \(SL_2(\mathbb R)\) [fcap-002I]

Definition. even subalgebra of Clifford algebra [wieser2022computing] [ca-001J]
CliffordAlgebra.even
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