Proposition. Real matrix blocks control Spin module complexification [fcap-0042]
✍️source

Let \(A=M_d(D)\), where \(d\geq 1\) and \(D\in \{\mathbb R,\mathbb C,\mathbb H\}\), regarded as a real algebra. Its unique irreducible real left module is \(U=D^d\), with commutant \(D^{\mathrm {op}}\) acting by right multiplication. Its complexification has the following form: \[\begin {array}{c|c|c|c} D&A\otimes _{\mathbb R}\mathbb C&U\otimes _{\mathbb R}\mathbb C &\dim _{\mathbb C}S\\\hline \mathbb R&M_d(\mathbb C)&S&d\\ \mathbb C&M_d(\mathbb C)\oplus M_d(\mathbb C)&S\oplus \overline S&d\\ \mathbb H&M_{2d}(\mathbb C)&S\oplus S&2d \end {array}\] In the complex row, the two summands have different central supports; in the quaternionic row, they are isomorphic. For a product of real matrix algebras, first select one simple block. These statements apply to irreducible real Spin modules obtained from even Clifford algebras because the actual Spin group spans the even algebra.

For the matrix-module assertion, the idempotents \(E_{ii}\) split any module into its coordinate corners, and \(E_{ij}\) identify the corners with one another. A simple module therefore has a simple \(D\)-module in its first corner, which is one-dimensional over the division algebra \(D\); this gives \(D^d\). For an endomorphism commuting with the full algebra, the matrix units make all coordinate maps the same. Commutation with the \(D\)-valued diagonal entries makes this common map \(f\) left \(D\)-linear, so \(f(x)=xf(1)\). It is therefore right multiplication by a single element on every coordinate. With \(R_a(u)=ua\), composition is \(R_aR_b=R_{ba}\), explaining the opposite algebra. Conjugation identifies \(D^{\mathrm {op}}\) with \(D\) when only its real algebra type is recorded. This is the real matrix-module classification of [lawson2016spin, Chapter I, Theorem 5.6, pp. 31--32]. The matrix-corner reconstruction has the general formal counterpart ModuleCat.matrixEquivalence, an equivalence between modules over a ring and modules over its matrix ring with a finite nonempty index type. The three real types and their commutants are the specializations calculated here.

The real row complexifies by entrywise scalar extension; conjugation is the resulting real structure. For the other rows, give \(U\) the complex structure \(u\mapsto ui\). The complexification of an underlying real complex space decomposes as \(S\oplus \overline S\), the two eigenspaces of this operator after scalar extension. For \(D=\mathbb C\), the central element \(i1\) acts as \(i\) on one summand and \(-i\) on the other. They are therefore inequivalent. For \(D=\mathbb H\), right multiplication \(J(u)=uj\) commutes with the left algebra action and obeys \(J(uz)=J(u)\overline z\) and \(J^2=-1\). Thus it identifies \(\overline S\) with \(S\). The quaternion matrix embedding Quaternion.toComplexMatrix_apply sends \(1,i,j,k\) to a complex basis of \(M_2(\mathbb C)\). Scalar extension consequently gives \(\mathbb H\otimes _{\mathbb R}\mathbb C\simeq M_2(\mathbb C)\) and then the displayed matrix algebra for general \(d\). Its action on \(S\) is the defining simple module of complex dimension \(2d\).

Here is the promised group-to-algebra bridge in every nondegenerate real signature. Choose orthogonal Clifford generators \(e_i\) with \(e_i^2=\varepsilon _i\in \{1,-1\}\). The real span \(L\) of Spin is a unital subalgebra, since products of linear combinations expand into linear combinations of group products. For \(i\ne j\) with \(\varepsilon _i\varepsilon _j=1\), \(e_ie_j\) itself lies in Spin by Remark [fcap-0028]. For opposite signs put \[v=\frac {5\varepsilon _i}{3}e_i+\frac 43e_j,\qquad Q(v)=\varepsilon _i,\qquad g=e_iv=\frac 53+\frac 43e_ie_j\in \operatorname {Spin}(Q).\] The last inclusion again uses \(Q(e_i)Q(v)=1\), hence the actual norm-one Lipschitz carrier. Therefore \(e_ie_j=\frac 34(g-\frac 53\,1)\in L\). All bivectors lie in \(L\); their products and \(1\) span the even Clifford algebra by its orthogonal monomial basis. Thus \[\operatorname {span}_{\mathbb R}\operatorname {Spin}(p,q)=\mathcal {C}\kern -2pt\ell ^+(p,q).\] It follows that real invariant subspaces and commuting endomorphisms for Spin are exactly those for the even algebra. TauCeti's CliffordAlgebra.span_spinGroup_eq_even_of_isSquare proves the spanning conclusion under a square-normalization hypothesis for every pair of nonisotropic vectors. That hypothesis holds for a definite real form, but fails for opposite-sign pairs. The displayed rational combination supplies the indefinite case separately.

For the compact positive form with \(n\geq 1\), the generators \(e_ie_n\), \(1\leq i<n\), square to \(-1\), anticommute, and generate the even algebra. Equality of dimensions gives \[\mathcal {C}\kern -2pt\ell ^+(n,0)\simeq \mathcal {C}\kern -2pt\ell (0,n-1).\] Combining the real classification Remark [fcap-0016] with the three rows above recovers the compact module types of Theorem [fcap-0041]. Lawson--Michelsohn write \(\mathcal {C}\kern -2pt\ell _n\) for our \(\mathcal {C}\kern -2pt\ell (0,n)\); their even-algebra identification and module table use this convention [lawson2016spin, Chapter I, (5.4)--(5.5) and Theorem 5.8, Table III, pp. 32--33]. Their compact Spin-spanning argument appears in Proposition 5.12, p. 36; the opposite-sign calculation above extends the linear-span argument to indefinite forms.

Signature still changes the real module type: \[\begin {aligned} \mathcal {C}\kern -2pt\ell ^+(2,2)&\simeq M_2(\mathbb R)\oplus M_2(\mathbb R),\\ \mathcal {C}\kern -2pt\ell ^+(4,0)&\simeq \mathbb H\oplus \mathbb H. \end {aligned}\] The split case gives real two-dimensional half-spin modules. The compact case gives quaternionic complex two-dimensional halves, each with real dimension four. The Lorentz even algebra \(M_2(\mathbb C)\) in Example [fcap-002L] has a real four-dimensional simple module; its complexification is the pair of inequivalent complex two-dimensional Weyl modules. Thus the compact Hermitian comparison cannot simply be reused for noncompact signatures. In all cases the scalar \(-1\in \operatorname {Spin}\) acts as \(-\operatorname {id}\), retaining the descent obstruction of Proposition [fcap-003M].

Proposition. Real matrix blocks control Spin module complexification [fcap-0042]
✍️source

Reference. Spin Geometry [lawson2016spin]

Polarization and the exterior spinor model [fcap-001G]

Definition. Polarization data and its carrier [chevalley1954algebraic, II.2.1--II.2.2, pp. 42--44] [fcap-001H]

SpinPolarizationData
L∃∀N

Convention. Clifford generators and polarized operators [fcap-0027]

Definition. The three polarized Clifford operators [fcap-001I]

wedge contract lineOperator cliffordOperator
L∃∀N

Lemma. The polarized Clifford relation [meinrenken2013clifford, Section 3.2.1, pp. 54--55] [fcap-001J]

contract_wedge cliffordOperator_sq
L∃∀N

Theorem. The exterior Clifford action [meinrenken2013clifford, Section 3.2.2, Proposition 3.5, p. 57] [fcap-001K]

spinAction
L∃∀N

Theorem. Creation and contraction generate all endomorphisms [meinrenken2013clifford, Section 3.2.4, Theorem 3.3, pp. 59--60] [fcap-001L]

creation_contraction_adjoin_eq_top spinAction_surjective
L∃∀N

Definition. The Spin and Pin representations [meinrenken2013clifford, Section 3.2.1, pp. 55--56] [fcap-001M]

spinRep pinRep
L∃∀N

Lemma. The restricted actions are the Clifford action [meinrenken2013clifford, Section 3.2.1, pp. 55--56] [fcap-001N]

spinRep_apply pinRep_apply
L∃∀N

Example. Rank one: matrix units and occupation [fcap-001P]

Theorem. Matrix units give the even-dimensional structure theorem [fcap-002V]

Proposition. An odd spinor action selects one Clifford summand [fcap-002W]

Theorem. Exterior pairings determine compact spinor reality [fcap-0041]

Proposition. Real matrix blocks control Spin module complexification [fcap-0042]
✍️source

Theorem. Occupation signs split the half-spin carrier [fcap-0025]

spinAction_diagonalBivector_basis iSup_spinWeightSpace_even_eq_spinPlus iSup_spinWeightSpace_odd_eq_spinMinus
L∃∀N

Proposition. The simple type-D operators generate the positive nilradical [fcap-002X]

Theorem. The two top exterior words are the type-D fork highest vectors [fcap-002Y]

Theorem. The top exterior word is the type-B spin highest vector [fcap-002Z]

Proposition. Exterior weights give the finite spin characters [fcap-0030]

Proposition. Complex Spin in dimension three is \(SL_2\) on the spin module [fcap-0031]

Proposition. Complex Spin in dimension four is two copies of \(SL_2\) [fcap-0032]

Proposition. Complex Spin in dimension five is the symplectic group on spinors [fcap-0033]

Proposition. Complex Spin in dimension six is \(SL_4\) on a half-spin module [fcap-0034]

Proposition. The outer nodes of \(D_4\) permute the three eight-dimensional classes [fcap-003B]

Theorem. The actual compact \(\operatorname {Spin}(8)\) has outer Lie-automorphism group \(S_3\) [fcap-003C]

Proposition. Clifford multiplication gives the triality trilinear [fcap-003D]

Theorem. A pinned three-cycle integrates to compact triality [fcap-003T]

Proof.

Theorem. Triality cycles three real modules and preserves the trilinear [fcap-003U]

Proof.

Remark. Horizons beyond the exterior model [meinrenken2013clifford, Section 3.2.4, Theorem 3.3, pp. 59--60]; [kostant1997clifford, Section 2.4, Theorem 8, pp. 286--287]; [lawson2016spin, I.2, pp. 13--19] [fcap-0017]

Theorem. Exterior pairings determine compact spinor reality [fcap-0041]

Theorem. Exterior pairings determine compact spinor reality [fcap-0041]

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

Remark. Reverse norm signs and the Spin carrier [fcap-0028]

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

Proposition. The central sign determines descent to the special orthogonal group [fcap-003M]

Proof.