Theorem. Reconstructing a skew Clifford lift in polar-dual coordinates [fcap-001Z]
✍️source
soEquivQuadratic_eq_sum_bivector
L∃∀N

    Let \(K\) be a field in which \(2\) is invertible, \(V\) a finite-dimensional \(K\)-space, and \(Q\) a nondegenerate quadratic form. Write \(B_Q\) for its unhalved polar form. For a basis \((b_i)\) let \((d_i)\) be its \(B_Q\)-dual basis, and let \(f\in \mathfrak {so}(V,B_Q)\). The quadratic Clifford element \(z_f=\operatorname {soEquivQuadratic}(f)\) has the coordinate expression \[z_f=\frac 12\sum _i\beta _Q\bigl (f(b_i),d_i\bigr ),\] where \(\beta _Q(u,v)=\tfrac 12(\iota (u)\iota (v)-\iota (v)\iota (u))\) is the bivector normalization of Convention [fcap-000N].

    [goodman2009symmetry, Definition 6.1.1(C1), p. 302] uses the complex Clifford relation \[\{\gamma (u),\gamma (v)\}=\beta (u,v)1.\] Thus their \(\beta \) is \(B_Q\), with \(Q(v)=\beta (v,v)/2\).

    [goodman2009symmetry, Section 6.2.1, Lemmas 6.2.1--2 and (6.13), pp. 312--313] gives the complex, nondegenerate basis-free construction through \(R_{a,b}(x)=\beta (b,x)a-\beta (a,x)b\) and its half-commutator lift. The displayed polar-dual sum is the coordinate reconstruction; Tau Ceti proves it over the stated wider class of fields.

    Proof. 

      For \(x\in V\), dual-basis reconstruction and skew-adjointness give the two contractions \[\begin {aligned} \sum _i B_Q(d_i,x)f(b_i)&=f(x),\\ \sum _i B_Q(f(b_i),x)d_i&=-f(x). \end {aligned}\] By Lemma [fcap-000P], the commutator of the sum of bivectors with \(\iota (x)\) is the image of their difference, hence \(2\iota (f(x))\). Multiplication by \(1/2\) gives the action of \(z_f\) on every generator. The nondegenerate quadratic realization in Theorem [fcap-000R] then identifies the two quadratic elements.

      References

        Reference. Symmetry, Representations, and Invariants [goodman2009symmetry]

        • 2009
        • Roe Goodman, Nolan R. Wallach
        @book{goodman2009symmetry,
 title = {Symmetry, Representations, and Invariants},
 author = {Goodman, Roe and Wallach, Nolan R.},
 year = {2009},
 series = {Graduate Texts in Mathematics},
 volume = {255},
 publisher = {Springer},
 doi = {10.1007/978-0-387-79852-3}
}

        Context

          Bivectors and the orthogonal Lie algebra [fcap-000M]

            The second exterior power singled out by the PBW filtration has a second life inside the Clifford algebra. Its half-commutators are closed under the commutator bracket, and their action on generators is exactly the infinitesimal orthogonal action. A polar-dual basis then reconstructs a general skew lift; the Killing form specializes that calculation to the adjoint action.

            § [ca-0001]

            Convention. Bivector normalizations [meinrenken2013clifford, Section 2.2.5, Proposition 2.5, p. 33] [fcap-000N]
            bivector_defbivector_lie_ι
            L∃∀N

              Meinrenken works throughout this chapter with finite-dimensional vector spaces over a characteristic-zero field. The cards below retain his normalization and formulas but follow TauCeti's extensions. Each card states the commutative-ring, module, invertible-\(2\), finite-dimensional, or nondegeneracy assumptions it needs.

              Meinrenken starts with a symmetric bilinear form \(B\) and the relation \[uv+vu=2B(u,v)1.\] For the associated quadratic form \(Q(v)=B(v,v)\), the polar form used here is \(B_Q=2B\).

              Meinrenken's quantization formula and the half-commutator agree: \[q(u\wedge v)=uv-B(u,v)1 =\frac 12(uv-vu)=:\beta (u,v).\] The induced action on a vector is \[[\beta (u,v),x] =B_Q(v,x)u-B_Q(u,x)v =2\bigl (B(v,x)u-B(u,x)v\bigr ).\] This is Meinrenken's formula \(-2\iota _{B(x,-)}(u\wedge v)\) and Kostant's formula \(-2\iota _x(u\wedge v)\) in the corresponding contraction convention [kostant1997clifford, Section 2.4, (12) and Theorem 8, pp. 283, 286]. The factor of two belongs to the passage from \(B\) to the unhalved polar form \(B_Q\); it is not an additional rescaling of \(\beta \).

              Definition. Clifford bivector [meinrenken2013clifford, Section 2.2.5, Proposition 2.5, p. 33] [fcap-000O]
              bivectorExteriorequivExterior_bivectorExterior
              L∃∀N

                For the formalized extension described in Convention [fcap-000N], let \(R\) be a commutative ring in which \(2\) is invertible, let \(M\) be an \(R\)-module, and let \(Q\) be a quadratic form on \(M\). The Clifford bivector of \(u,v\in M\) is \[\beta _Q(u,v)=\frac 12\bigl (\iota (u)\iota (v)-\iota (v)\iota (u)\bigr ).\] It is alternating in \(u\) and \(v\), so it induces a linear map \[\beta _Q:\bigwedge _R^2M\longrightarrow \mathcal {C}\kern -2pt\ell (Q).\] The exterior model is a left inverse on its image: \[\operatorname {equivExterior}_Q\bigl (\beta _Q(z)\bigr )=z \qquad (z\in \bigwedge _R^2M).\] Hence \(\beta _Q\) is injective.

                Lemma. A bivector acts by an infinitesimal rotation [meinrenken2013clifford, Section 2.2.10, Theorem 2.1, pp. 39--40] [fcap-000P]
                bivector_lie_ι
                L∃∀N

                  For \(u,v,x\in M\), the commutator of the bivector with a Clifford generator is \[[\beta _Q(u,v),\iota (x)] =\iota \left (B_Q(v,x)u-B_Q(u,x)v\right ).\] The endomorphism \[A_{u\wedge v}(x)=B_Q(v,x)u-B_Q(u,x)v\] is skew-adjoint for \(B_Q\), since \[B_Q(A_{u\wedge v}x,y)+B_Q(x,A_{u\wedge v}y)=0.\] Thus commutation by a degree-two Clifford element preserves the generating module and realizes the elementary infinitesimal orthogonal transformation attached to \(u\wedge v\).

                  Proof. 

                    Expand the half-commutator and move \(\iota (x)\) past \(\iota (u)\) and \(\iota (v)\) using the Clifford relation. The cubic terms cancel; the two polar terms combine to the displayed generator. Symmetry of \(B_Q\) then makes the skew-adjointness equation cancel in pairs.

                    Theorem. Exterior bivectors are the quadratic Clifford Lie algebra [meinrenken2013clifford, Section 2.2.10, Theorem 2.1, pp. 39--40] [fcap-000Q]
                    bivectorExteriorEquivQuadraticLieSubalgebrabivectorLieEquiv
                    L∃∀N

                      Over the commutative ring and module of Definition [fcap-000O], let \(\mathfrak {cl}^{(2)}(Q)\) be the submodule of \(\mathcal {C}\kern -2pt\ell (Q)\) spanned by the elements \(\beta _Q(u,v)\). This is TauCeti's module-general extension of Meinrenken's characteristic-zero vector-space construction. The commutator formula of Lemma [fcap-000P], together with the Jacobi identity, shows that this submodule is closed under commutators. Transporting its bracket across \(\beta _Q\) gives a Lie algebra structure on \(\bigwedge _R^2M\) and a Lie equivalence \[\bigwedge _R^2M\simeq _{\mathrm {Lie}}\mathfrak {cl}^{(2)}(Q).\] On a decomposable bivector the equivalence is exactly \[u\wedge v\longmapsto \frac 12(uv-vu).\] No choice of basis enters this identification.

                      Theorem. Quadratic elements realize the orthogonal Lie algebra [meinrenken2013clifford, Section 2.2.10, Proposition 2.12, p. 40] [fcap-000R]
                      soEquivQuadraticsoEquivQuadratic_lie_ι
                      L∃∀N

                        Meinrenken states this realization over a characteristic-zero field. TauCeti proves the extension below over any field of characteristic different from \(2\). Let \(K\) be such a field, let \(V\) be finite-dimensional, and let \(Q\) be nondegenerate. Write \(\mathfrak {so}(V,B_Q)\) for the Lie algebra of endomorphisms skew-adjoint for the polar form. Then commutation on generators gives a Lie equivalence \[\mathfrak {so}(V,B_Q)\simeq _{\mathrm {Lie}}\mathfrak {cl}^{(2)}(Q).\] If \(A\in \mathfrak {so}(V,B_Q)\) corresponds to \(z_A\in \mathfrak {cl}^{(2)}(Q)\), then \[[z_A,\iota (x)]=\iota (Ax)\qquad (x\in V).\] Combining this equivalence with Theorem [fcap-000Q] gives the basis-free chain \[\bigwedge _K^2V\simeq _{\mathrm {Lie}}\mathfrak {so}(V,B_Q) \simeq _{\mathrm {Lie}}\mathfrak {cl}^{(2)}(Q).\] Kostant identifies the same degree-two subspace with \(\operatorname {Lie}(\operatorname {Spin}(V))\) and the first equivalence with the differential of the double cover [kostant1997clifford, Section 2.4, Theorem 8, pp. 286--287].

                        Example. An elementary skew matrix [meinrenken2013clifford, Section 2.2.10, Proposition 2.12, p. 40] [fcap-000S]
                        bivectorEquivSo_apply_ιMultibivectorEquivSo_apply_ιMulti_mulVec
                        L∃∀N

                          This coordinate example specializes TauCeti's extension of the characteristic-zero construction summarized in Convention [fcap-000N]. Take \(V=R^n\) with the standard sum-of-squares quadratic form, and let \(e_i,e_j\) be distinct coordinate vectors. Under the standard-coordinate realization, \[e_i\wedge e_j\longmapsto 2(E_{ij}-E_{ji}).\] Thus its action on \(x=(x_k)\) is \[x\longmapsto 2x_j e_i-2x_i e_j.\] The factor \(2\) is the polar-form normalization from Convention [fcap-000N]: for the sum-of-squares form, \(B_Q(x,y)=2\sum _kx_ky_k\). The displayed matrix is the standard-coordinate specialization of the basis-free isomorphism in Meinrenken's proposition. Dividing the skew matrix by two would correspond to using the associated bilinear form \(B_Q/2\) instead.

                          Theorem. Reconstructing a skew Clifford lift in polar-dual coordinates [fcap-001Z]
                          ✍️source
                          soEquivQuadratic_eq_sum_bivector
                          L∃∀N

                            Let \(K\) be a field in which \(2\) is invertible, \(V\) a finite-dimensional \(K\)-space, and \(Q\) a nondegenerate quadratic form. Write \(B_Q\) for its unhalved polar form. For a basis \((b_i)\) let \((d_i)\) be its \(B_Q\)-dual basis, and let \(f\in \mathfrak {so}(V,B_Q)\). The quadratic Clifford element \(z_f=\operatorname {soEquivQuadratic}(f)\) has the coordinate expression \[z_f=\frac 12\sum _i\beta _Q\bigl (f(b_i),d_i\bigr ),\] where \(\beta _Q(u,v)=\tfrac 12(\iota (u)\iota (v)-\iota (v)\iota (u))\) is the bivector normalization of Convention [fcap-000N].

                            [goodman2009symmetry, Definition 6.1.1(C1), p. 302] uses the complex Clifford relation \[\{\gamma (u),\gamma (v)\}=\beta (u,v)1.\] Thus their \(\beta \) is \(B_Q\), with \(Q(v)=\beta (v,v)/2\).

                            [goodman2009symmetry, Section 6.2.1, Lemmas 6.2.1--2 and (6.13), pp. 312--313] gives the complex, nondegenerate basis-free construction through \(R_{a,b}(x)=\beta (b,x)a-\beta (a,x)b\) and its half-commutator lift. The displayed polar-dual sum is the coordinate reconstruction; Tau Ceti proves it over the stated wider class of fields.

                            Proof. 

                              For \(x\in V\), dual-basis reconstruction and skew-adjointness give the two contractions \[\begin {aligned} \sum _i B_Q(d_i,x)f(b_i)&=f(x),\\ \sum _i B_Q(f(b_i),x)d_i&=-f(x). \end {aligned}\] By Lemma [fcap-000P], the commutator of the sum of bivectors with \(\iota (x)\) is the image of their difference, hence \(2\iota (f(x))\). Multiplication by \(1/2\) gives the action of \(z_f\) on every generator. The nondegenerate quadratic realization in Theorem [fcap-000R] then identifies the two quadratic elements.

                              Theorem. The Killing-adjoint lift has coefficient one quarter [fcap-0020]
                              adjointCliffordHom_eq_sum_bivector
                              L∃∀N

                                Let \(K\) be a field in which \(2\) is invertible, and let \(L\) be a finite-dimensional \(K\)-Lie algebra whose Killing form \(\kappa \) is nondegenerate (the formal hypothesis is \(\operatorname {IsKilling}_K(L)\)). Set \(Q_\kappa (y)=\kappa (y,y)\). If \((b_i)\) is a basis and \((b_i^\vee )\) its Killing-dual basis, then for \(x\in L\) the quadratic Clifford lift of the adjoint action is \[\operatorname {adjointCliffordHom}(x) =\frac 14\sum _i\beta _{Q_\kappa } \bigl ([x,b_i],b_i^\vee \bigr ).\] Here \(\beta _{Q_\kappa }\) is the half-commutator bivector of Convention [fcap-000N]; the summand is the value of CliffordAlgebra.adjointBivector_apply.

                                Kostant identifies the complex orthogonal Lie algebra with \(\bigwedge ^2L\) and constructs the adjoint map from a complex semisimple \(L\) using its Killing pairing [kostant1997clifford, Sections 1.1--1.2, pp. 275--276]. This supports the complex adjoint construction. The coordinate expression and its \(1/4\) normalization are the calculation here; Tau Ceti's theorem uses the stated \(\operatorname {IsKilling}\) hypothesis over a wider field class.

                                Proof. 

                                  The invariance of \(\kappa \) makes \(\operatorname {ad}_x\) skew-adjoint. Applying Theorem [fcap-001Z] to \(f=\operatorname {ad}_x\) first gives a factor \(1/2\) with the dual basis for \(B_{Q_\kappa }\). Since \(B_{Q_\kappa }=2\kappa \), that polar-dual basis is \(d_i=\tfrac 12 b_i^\vee \). Bilinearity of the bivector gives the second factor \(1/2\), and \((1/2)(1/2)=1/4\). The equality \(B_{Q_\kappa }=2\kappa \) is the convention encoded by TauCeti.LieAlgebra.polarBilin_killingQuadraticForm.

                                  Remark. Infinitesimal and global Clifford symmetry [fcap-000T]

                                    The quadratic Clifford Lie algebra is the infinitesimal part of the same conjugation action that produces reflections. Kostant proves that \(\bigwedge ^2V=\operatorname {Lie}(\operatorname {Spin}(V))\), that its map to \(\mathfrak {so}(V)\) is the differential of \(\operatorname {Spin}(V)\to SO(V)\), and that its commutator action extends as the corresponding derivation of the exterior algebra [kostant1997clifford, Section 2.4, Theorem 8, pp. 286--287]. Chevalley obtains the same two-vector Lie algebra inside the Clifford group [chevalley1954algebraic, II.2.9, pp. 67--68].

                                    The cards above identify the algebraic Lie layer and reconstruct its lifts in coordinates. They do not construct Lie-group structures or identify a differential of a Lie-group covering map. The module-action card that follows uses the quadratic lift; the global orthogonal group is reached elsewhere by finite products of reflections.

                                    Theorem. An orthogonal Lie action lifts to every Clifford module [kostant1997clifford, Section 3.1, pp. 294--295] [fcap-001O]
                                    cliffordInducedRepcliffordInducedRep_apply
                                    L∃∀N

                                      Let \(K\) be a field in which \(2\) is invertible, let \(V\) be finite-dimensional, and let \(Q\) be nondegenerate. Suppose a \(K\)-Lie algebra \(L\) acts orthogonally on \(V\) through \[\theta :L\longrightarrow \mathfrak {so}(V,B_Q),\] and let \(\rho :\mathcal {C}\kern -2pt\ell (Q)\to \operatorname {End}_K(S)\) be any Clifford module. The equivalence of Theorem [fcap-000R] lifts \(\theta \) to quadratic Clifford elements; composing with \(\rho \) gives a Lie representation \[\rho _{\mathcal {C}\kern -2pt\ell }:L\longrightarrow \operatorname {End}_K(S),\qquad \rho _{\mathcal {C}\kern -2pt\ell }(y)=\rho \bigl (\operatorname {soEquivQuadratic}(\theta (y))\bigr ).\] Kostant constructs this lift for a complex reductive group representation and then lets its quadratic elements act on the spin module. The formalized statement isolates the algebraic mechanism over the stated field and for an arbitrary Clifford module; it does not assert that \(\theta \) is a differential of a group representation.

                                      Proposition. Left Clifford multiplication is not the commutator action [fcap-003E]

                                        Let \(\mathfrak g\) be a finite-dimensional complex semisimple Lie algebra with Killing form \(\kappa \), and put \(Q_\kappa (x)=\kappa (x,x)\). If \((b_i)\) is a basis and \((b_i^\vee )\) its Killing-dual basis, the normalized quadratic lift of the adjoint action is \[\delta (x)=\frac 14\sum _i \beta _{Q_\kappa }([x,b_i],b_i^\vee )\in \mathcal {C}\kern -2pt\ell (Q_\kappa )^0.\] The coefficient is the one derived in Theorem [fcap-0020]: the polar form is \(B_{Q_\kappa }=2\kappa \), so the polar-dual basis contributes one factor \(1/2\), and the general lift contributes the other. This is a grade-two element, not the canonical generator \(\iota _{Q_\kappa }(x)\).

                                        There are two natural actions on the same Clifford vector space: \[L_x(c)=\delta (x)c,\qquad D_x(c)=\delta (x)c-c\delta (x).\] If \(R_a(c)=ca\), then \[D_x=L_x-R_{\delta (x)},\qquad [L_x,R_a]=0.\] Under equivariant Clifford antisymmetrization, \(D\) is the derivation extension of the adjoint action to \(\bigwedge \mathfrak g\). The action \(L\) is instead left multiplication on the Clifford algebra. On matrix coordinates \(\operatorname {End}(S)\simeq S\otimes S^*\), it acts on the \(S\) factor and leaves \(S^*\) as a multiplicity space.

                                        TauCeti defines the selected left action and proves these comparisons in CliffordAlgebra.kostantLieModule CliffordAlgebra.kostant_lie_def CliffordAlgebra.kostant_lie_sub_mul CliffordAlgebra.kostantRightMul. The generic module lift in Theorem [fcap-001O] supplies the same algebraic mechanism, but it does not identify the left action with the exterior derivation action.

                                        [kostant1997clifford, Sections 3.1 and 5.2, pp. 294 and 311] defines the left action through this quarter-normalized lift and separately identifies the commutator with the exterior extension. [panyushev2001exterior, Section 2, internal preprint pp. 7--8] studies the exterior/spin action under the same complex semisimple hypotheses. The following example shows why their roles cannot be exchanged.

                                        Example. For \(\mathfrak {sl}_2\), the two Clifford actions have different constituents [fcap-003F]

                                          Take the basis \(e,f,h\) of \(\mathfrak {sl}_2(\mathbb C)\) with \[[h,e]=2e,\qquad [h,f]=-2f,\qquad [e,f]=h.\] The Killing form satisfies \(\kappa (e,f)=4\) and \(\kappa (h,h)=8\). Hence the Clifford relations are \[e^2=f^2=0,\qquad ef+fe=8,\qquad h^2=8,\qquad he=-eh,\quad hf=-fh.\] Substitution in the lift of Proposition [fcap-003E] gives \[E=\delta (e)=\frac {he}{8},\qquad F=\delta (f)=\frac {fh}{8},\qquad H=\delta (h)=\frac {ef-fe}{8}=\frac {ef}{4}-1.\] Direct Clifford multiplication yields \[E^2=F^2=0,\qquad H^2=1,\qquad EF=\frac {1+H}{2},\qquad FE=\frac {1-H}{2}.\] Thus \(E,F,EF,FE\) are the four matrix units of the lifted \(\mathfrak {sl}_2\) action.

                                          For left multiplication, the \(H=+1\) and \(H=-1\) subspaces are interchanged by \(F\) and \(E\); the two maps are inverse on the corresponding projected spaces because \(EF\) and \(FE\) are their identity projectors. They have equal dimension, and \(\dim \mathcal {C}\kern -2pt\ell (Q_\kappa )=8\), so each has dimension four. Choosing a basis of the positive space and its \(F\)-images displays \[(\mathcal {C}\kern -2pt\ell (Q_\kappa ),L)\simeq 4V_1,\] four copies of the standard two-dimensional module.

                                          For the commutator action \(D\), antisymmetrization identifies the same vector space with \(\bigwedge \mathfrak {sl}_2\). The exterior powers are \[\bigwedge ^0\mathfrak g\simeq V_0,\qquad \bigwedge ^1\mathfrak g\simeq V_2,\qquad \bigwedge ^2\mathfrak g\simeq V_2,\qquad \bigwedge ^3\mathfrak g\simeq V_0,\] where the middle identification uses the bracket and the last uses the invariant volume. Therefore \[(\mathcal {C}\kern -2pt\ell (Q_\kappa ),D)\simeq 2V_0\oplus 2V_2.\] The dimensions agree but the characters do not. [kostant1997clifford, Theorem 39, pp. 312--313] gives the first answer through Kostant's primary-type theorem. [panyushev2001exterior, Example 2.5(1), internal preprint p. 8] independently gives the exterior decomposition. The displayed matrix-unit calculation directly verifies which action produces which answer.

                                          Theorem. The left Killing-Clifford module is primary of type \(V_\rho \) [fcap-003G]

                                            Let \(\mathfrak g\) be a finite-dimensional complex semisimple Lie algebra of rank \(\ell \). Choose positive roots \(\Phi ^+\), put \(r=\lvert \Phi ^+\rvert \), and write \(\rho =\frac 12\sum _{\alpha >0}\alpha \). Then \[\dim \mathfrak g=\ell +2r.\] Give \(\mathcal {C}\kern -2pt\ell (Q_\kappa )\) the left action \(L\) of Proposition [fcap-003E]. As a \(\mathfrak g\)-module it is \[\mathcal {C}\kern -2pt\ell (Q_\kappa )\simeq 2^{\ell +r}V_\rho .\] Here the exponent is the multiplicity; \(\dim V_\rho =2^r\), so the total dimension is \(2^{\ell +2r}\), as required. For the zero Lie algebra this reads as the one-dimensional trivial module.

                                            First compute the spinor constituent. The weights of the adjoint orthogonal module are zero with multiplicity \(\ell \) and the roots in opposite pairs. The exterior spinor character is therefore \[\operatorname {ch}S= 2^{\lfloor \ell/2\rfloor } \prod _{\alpha >0} \left (e^{\alpha/2}+e^{-\alpha/2}\right ).\] The Weyl character formula for highest weight \(\rho \), together with the Weyl denominator identity applied to \(2\rho \) and divided by the ordinary denominator, gives \[\operatorname {ch}V_\rho =e^\rho \prod _{\alpha >0}(1+e^{-\alpha }) =\prod _{\alpha >0} \left (e^{\alpha/2}+e^{-\alpha/2}\right ).\] Complete reducibility and independence of irreducible characters now give \[S\simeq 2^{\lfloor \ell/2\rfloor }V_\rho .\] [panyushev2001exterior, Example 2.5(1), internal preprint p. 8] records this adjoint specialization under the complex semisimple hypotheses. The character calculation above supplies the comparison in the fixed normalization rather than substituting that conclusion without proof.

                                            It remains to pass from a spinor carrier to the full Clifford algebra. If \(n=\ell +2r\) is even, the matrix range of Theorem [fcap-002V] gives \[\mathcal {C}\kern -2pt\ell (Q_\kappa )\simeq \operatorname {End}(S)\simeq S\otimes S^*.\] Left multiplication acts on the first factor, so \(S^*\) is a trivial multiplicity space of dimension \(2^{n/2}\). If \(n\) is odd, the odd structure of Proposition [fcap-002W] gives two matrix blocks \(\operatorname {End}(S_1)\oplus \operatorname {End}(S_2)\). The two full Clifford actions differ, but their restrictions to the even algebra are equivalent; since \(\delta (\mathfrak g)\) is even, both blocks contain the same \(\mathfrak g\)-module. Their combined multiplicity-space dimension is \(2\cdot 2^{(n-1)/2}=2^{\lceil n/2\rceil }\).

                                            In both parity cases the multiplicity of \(V_\rho \) is \[2^{\lceil n/2\rceil } 2^{\lfloor \ell/2\rfloor } =2^{\ell +r}.\] This proves the displayed decomposition. [kostant1997clifford, Theorem 39, pp. 312--313] independently proves the exact left-regular primary-type statement by a highest-weight right ideal. TauCeti formalizes the selected left action and its commuting right multiplications, but the character, matrix-block, and multiplicity synthesis here is not a packaged formal theorem.

                                            The conclusion is a module decomposition only. It does not assert the stronger algebra factorization \(\operatorname {End}(V_\rho )\otimes \mathcal {C}\kern -2pt\ell (P)\), identify the full commutant, or invoke the Hopf--Koszul--Samelson primitive algebra. The normal-ordered \(\mathfrak {gl}_N\) calculation in the separate CAR section uses a trace shift and is not a substitute for this Killing-adjoint theorem.

                                            Theorem. The invariant algebra is the internal commutant [fcap-003P]

                                              Let \(\mathfrak g\) be complex semisimple of rank \(\ell \), with \(r\) positive roots and \(\rho \) their half-sum. Put \(A=\mathcal {C}\kern -2pt\ell (Q_\kappa )\), where \(Q_\kappa (x)=\kappa (x,x)\), and use the lift \(\delta \) of Proposition [fcap-003E]. Define the unital algebra \(E=\delta (U(\mathfrak g))\subseteq A^0\) and the commutator invariants \[J=\{a\in A:[\delta (x),a]=0\text { for every }x\in \mathfrak g\}.\] Then \(J=\operatorname {Cent}_A(E)\), \(E\simeq \operatorname {End}(V_\rho )\), and multiplication is an isomorphism of ordinary complex algebras \[E\otimes J\longrightarrow A,\qquad e\otimes j\longmapsto ej.\] In particular, \(\dim J=2^\ell \). These assertions include the zero Lie algebra, where \(A=E=J=\mathbb C\).

                                              An element commutes with the generators \(\delta (\mathfrak g)\) exactly when it commutes with their unital associative algebra, proving the centralizer assertion. Set \(d=2^{\lfloor \ell /2\rfloor }\). The spinor calculation in Theorem [fcap-003G] gives \(S\simeq V_\rho \otimes \mathbb C^d\). The associative algebra generated by an irreducible complex representation is its full endomorphism algebra (Burnside's theorem). Thus the image of \(U(\mathfrak g)\) on \(S\) is \(\operatorname {End}(V_\rho )\otimes 1\). This is the primary-representation calculation in [kostant1997clifford, equations (74)--(76), p. 295].

                                              If \(\ell \) is even, then \(\dim \mathfrak g=\ell +2r\) is even, and Theorem [fcap-002V] identifies \(A\) with \(\operatorname {End}(S)\). Matrix units in \(\operatorname {End}(V_\rho )\) show that its commutant on \(V_\rho \otimes \mathbb C^d\) is \(1\otimes \operatorname {End}(\mathbb C^d)\): commuting with diagonal units forces a block-diagonal operator, and commuting with off-diagonal units makes all its diagonal blocks equal. Consequently \[J\simeq \operatorname {End}(\mathbb C^d),\qquad E\otimes J\simeq \operatorname {End}(V_\rho \otimes \mathbb C^d)=A.\] This also gives \(\dim J=d^2=2^\ell \).

                                              If \(\ell \) is odd, Proposition [fcap-002W] identifies \(A\) with \(\operatorname {End}(S_+)\oplus \operatorname {End}(S_-)\). The two spinor restrictions to \(A^0\) are equivalent and faithful. After identifying them, \(E\subseteq A^0\) is the diagonal copy of \(\operatorname {End}(V_\rho )\otimes 1\). The same matrix-unit argument in each block gives \[J\simeq \operatorname {End}(\mathbb C^d) \oplus \operatorname {End}(\mathbb C^d).\] Multiplication \(E\otimes J\to A\) is the preceding matrix isomorphism in each of the two summands, and \(\dim J=2d^2=2^\ell \). This proves both parity cases without assuming that the full odd-dimensional Clifford algebra is simple. [kostant1997clifford, Propositions 17--20, pp. 294--297] gives this centralizer factorization for any primary spin representation.

                                              Under the commutator action, \(\mathfrak g\) acts trivially on \(J\) and by conjugation on \(E\). Equivariant antisymmetrization from Proposition [fcap-003E] therefore yields the exterior-module consequence \[\bigwedge \mathfrak g\simeq 2^\ell (V_\rho \otimes V_\rho ^*).\] This multiplicity concerns the commutator action; the left-action multiplicity in Theorem [fcap-003G] is \(2^{\ell +r}\). The algebra factorization and its proof here are not a packaged formal theorem.

                                              Remark. Three different commutants for the left Clifford action [fcap-003Q]

                                                Keep the hypotheses and notation of Theorem [fcap-003P], and let \(L_x(c)=\delta (x)c\). The internal centralizer \(J\subseteq A\), the right-multiplication operators on \(A\), and the full algebra of \(\mathfrak g\)-module endomorphisms of \((A,L)\) have distinct roles.

                                                Left multiplication by \(j\in J\) commutes with every \(L_x\). Right multiplication by any \(a\in A\) also commutes with every \(L_x\), purely by associativity. Thus there are injective algebra maps \[J\longrightarrow \operatorname {End}_{\mathfrak g}(A,L), \qquad j\longmapsto (c\longmapsto jc),\] \[A^{\mathrm {op}}\longrightarrow \operatorname {End}_{\mathfrak g}(A,L), \qquad a\longmapsto (c\longmapsto ca).\] Evaluation at \(1\) proves injectivity. The opposite algebra in the second map records the reversed order of right compositions.

                                                By Theorem [fcap-003G], \((A,L)\simeq V_\rho \otimes \mathbb C^m\) with \(m=2^{\ell +r}\). Schur's lemma and the same block calculation as in Theorem [fcap-003P] give \[\operatorname {End}_{\mathfrak g}(A,L) \simeq M_m(\mathbb C).\] The dimensions of the three algebras are therefore \[\dim J=2^\ell ,\qquad \dim A^{\mathrm {op}}=2^{\ell +2r},\qquad \dim \operatorname {End}_{\mathfrak g}(A,L)=2^{2\ell +2r}.\] For \(\mathfrak g=\mathfrak {sl}_2\), these are \(2\), \(8\), and \(16\), respectively: \(J\simeq \mathbb C\oplus \mathbb C\), while \(\operatorname {End}_{\mathfrak g}(A,L)\simeq M_4(\mathbb C)\). The four-copy left-module calculation is explicit in Example [fcap-003F]. For every nonzero complex semisimple \(\mathfrak g\), the dimensions show that right multiplications form a proper subalgebra of the full module commutant. They do not by themselves identify that commutant.

                                                Theorem. Primitive invariants and their quadratic form [fcap-003R]

                                                  Definition [ca-000J]

                                                  Let \(\mathfrak g\) be finite-dimensional complex semisimple of rank \(\ell \). Write \(I=(\bigwedge \mathfrak g)^{\mathfrak g}\) for the adjoint invariants and \(I^+\) for their positive-degree part. Extend the Killing form \(\kappa \) to the determinant pairing \(\langle -,-\rangle \) on each exterior degree, with different degrees orthogonal. Let \(\pi :\bigwedge \mathfrak g\to I\) be the invariant projection along \(\mathfrak g\mathbin {\cdot }\bigwedge \mathfrak g\), the span of all adjoint derivatives. The coproduct on \(I\) is \[\Delta =(\pi \otimes \pi )\Delta _0|_I,\] where \(\Delta _0(x)=x\otimes 1+1\otimes x\) for \(x\in \mathfrak g\). It is this projected coproduct that defines primitive invariants: \[P=\{p\in I:\Delta p=p\otimes 1+1\otimes p\} =I^+\cap ((I^+)^2)^\perp .\] The ordinary exterior coproduct need not take \(I\) into \(I\otimes I\). The Lie-homology construction makes \(I\), with the displayed \(\Delta \), a graded super Hopf algebra; multiplication on \(I\otimes I\) uses the Koszul rule \((a\otimes b)(c\otimes d)=(-1)^{\lvert b\rvert \lvert c\rvert }ac\otimes bd\) for homogeneous elements. The coproduct is dual to the wedge product. This is the input from [meinrenken2013clifford, Theorem 10.1 and Proposition 10.11, pp. 240--241].

                                                  The Hopf--Koszul--Samelson theorem says that inclusion extends to \[\bigwedge P\xrightarrow {\sim }I.\] Here every primitive has odd exterior degree. Indeed, a nonzero homogeneous even primitive \(p\) would have a least nilpotent power \(p^k=0\); the component \(k p\otimes p^{k-1}\) of \(\Delta (p^k)\) could not vanish in characteristic zero. The orthogonal decomposition \(I^+=P\oplus (I^+)^2\) then proves generation: substituting it repeatedly expresses every element as a polynomial in \(P\), since the augmentation ideal is nilpotent. For independence, contraction by a primitive acts as an odd derivation on \(I\), and contraction of two primitives is a scalar. The first assertion is dual to \(\Delta \) preserving multiplication; the second follows from \(P\perp (I^+)^2\). Successive contractions by a dual primitive basis extract the coefficients of an exterior-monomial relation, starting with the largest number of factors. All coefficients vanish. This proves the stated isomorphism; compare [meinrenken2013clifford, Theorem 10.2 and Proposition 10.13, pp. 242--243]. By Theorem [fcap-003P], \(\dim I=2^\ell \), so \(\dim P=\ell \).

                                                  For \(x\in \mathfrak g\), let \(\iota _x\) be contraction using \(\kappa \), and set \(\iota _{x_1\wedge \cdots \wedge x_k}=\iota _{x_1}\cdots \iota _{x_k}\). Exterior reversion is \(\operatorname {rev}(u)=(-1)^{k(k-1)/2}u\) on degree \(k\). The form relevant to Clifford multiplication is \[B_\wedge (u,v)=(\iota _u v)_0 =\langle \operatorname {rev}(u),v\rangle ,\qquad Q_P(p)=B_\wedge (p,p).\] In particular, on primitive degree \(2m+1\), \(B_\wedge (p,q)=(-1)^m\langle p,q\rangle \). The Clifford polar relation for \(Q_P\) is therefore \(pq+qp=2B_\wedge (p,q)\), with the same generator-square convention as \(Q_\kappa (x)=\kappa (x,x)\).

                                                  This restriction of \(B_\wedge \) to \(P\) is nondegenerate. Choose a compact real form. Its Killing form is negative definite; on exterior degree \(k\), the determinant pairing is \((-1)^k\) times a positive definite real form. The subspace \(P\) is graded and preserved by compact conjugation, so the determinant pairing and its degreewise sign multiple \(B_\wedge \) are nondegenerate there. This also explains the normalization change from [meinrenken2013clifford, Section 11.3, p. 260], which starts with a positive compact form. Replacing that form by its negative preserves the graded orthogonal subspaces while changing the indicated degree signs.

                                                  Theorem. Kostant's primitive Clifford factorization [fcap-003S]

                                                    With \(P,Q_P\) as in Theorem [fcap-003R], put \(A=\mathcal {C}\kern -2pt\ell (Q_\kappa )\) and \(J=A^{\mathfrak g}\) for the commutator invariants of Theorem [fcap-003P]. Let \(q:\bigwedge \mathfrak g\to A\) be antisymmetrization. Then \(p\mapsto q(p)\) extends to an isomorphism of complex superalgebras \[\mathcal {C}\kern -2pt\ell (Q_P)\xrightarrow {\sim }J.\] Consequently multiplication gives \[\operatorname {End}(V_\rho )\otimes \mathcal {C}\kern -2pt\ell (Q_P) \xrightarrow {\sim }\mathcal {C}\kern -2pt\ell (Q_\kappa ).\] The endomorphism factor is purely even. Thus the displayed tensor product has the ordinary multiplication as well as the superalgebra interpretation. The proof below establishes the Clifford relation and generation separately.

                                                    We use the Lie-homology Hodge decomposition \[\bigwedge \mathfrak g =I\oplus \operatorname {im}d\oplus \operatorname {im}\partial =I\oplus (\mathfrak g\mathbin {\cdot }\bigwedge \mathfrak g),\] orthogonal for \(B_\wedge \). Here \(\partial \) is the Lie-homology boundary and \(d\) is the cohomology differential transported by \(\kappa \). The needed contraction identities are \[\iota _{\operatorname {im}d}I\subseteq \operatorname {im}\partial , \qquad \iota _{\operatorname {im}\partial }I\subseteq \operatorname {im}d,\] \[\ker \partial \wedge \operatorname {im}\partial \subseteq \mathfrak g\mathbin {\cdot }\bigwedge \mathfrak g.\] Proposition 10.8 and Lemmas 10.4--10.5 of [meinrenken2013clifford, pp. 238--239] give these identities. Changing the compact form's sign multiplies contractions and the transported differential by degreewise nonzero scalars, so the displayed ranges and identities also hold for \(\kappa \).

                                                    Choose a homogeneous basis \(\tau _a\) of the exterior algebra adapted to the three Hodge summands, and its \(B_\wedge \)-dual basis \(\tau ^a\). For odd \(p\in P\) and \(q'\in P\), the Clifford product formula is \[q^{-1}(q(p)q(q'))= \sum _a\iota _{\tau _a}p\wedge \iota _{\tau ^a}q'.\] This is the contraction expansion in [meinrenken2013clifford, equation (11.10), p. 260]; using the \(B_\wedge \)-dual basis incorporates the reversion signs. Terms indexed by \(\operatorname {im}d\) lie in \(\mathfrak g\mathbin {\cdot }\bigwedge \mathfrak g\) by the two contraction identities and the boundary-product identity. Terms indexed by \(\operatorname {im}\partial \) lie in \(\operatorname {im}d\), because \(d\) is a derivation and the wedge product of two exact elements is exact. Their combined sum is zero: the full product and the terms indexed by \(I\) are invariant, whereas the remaining terms lie in the complementary Hodge summand.

                                                    Take a homogeneous basis \(p_1,\ldots ,p_\ell \) of \(P\) and its \(B_\wedge \)-dual basis \(p^1,\ldots ,p^\ell \). The HKS monomials give a basis of \(I\). Repeated primitive contractions of a primitive vanish after the first contraction, which is already scalar. Moreover, \(\mathbb C\oplus P\oplus (I^+)^2\) is orthogonal, so the dual vectors belonging to the single primitives are \(p^i\). Only the constant monomial and the single primitives survive in the preceding sum. Hence \[q^{-1}(q(p)q(q')) =p\wedge q'+\sum _i B_\wedge (p_i,p)B_\wedge (p^i,q') =p\wedge q'+B_\wedge (p,q').\] Both primitives are odd, so swapping them cancels their wedge products and yields \[q(p)q(q')+q(q')q(p)=2B_\wedge (p,q').\] This is Kostant's primitive Clifford relation in the fixed sign convention; see [meinrenken2013clifford, Theorem 11.6 and equation (11.13), pp. 260--261] and [kostant1997clifford, Theorem 35, pp. 306--308].

                                                    The universal property now gives a homomorphism \(\mathcal {C}\kern -2pt\ell (Q_P)\to J\). To prove generation, order the HKS monomials by their total exterior degree. The product \(q(p_{i_1})\cdots q(p_{i_s})\) has leading exterior term \(p_{i_1}\wedge \cdots \wedge p_{i_s}\) under \(q^{-1}\); its other terms have lower degree, by the Clifford filtration. They are invariant, hence linear combinations of lower-degree HKS monomials. Induction therefore puts every \(q(p_{i_1}\wedge \cdots \wedge p_{i_s})\) in the generated algebra. This proves surjectivity. Both algebras have dimension \(2^\ell \), so the map is an isomorphism. It preserves parity, since all elements of \(P\) have odd degree.

                                                    Finally, substitute this identification of \(J\) in the multiplication isomorphism of Theorem [fcap-003P]. This yields the claimed factorization, also displayed in [meinrenken2013clifford, equation (11.14), p. 262]. For \(\mathfrak {sl}_2\), \(P\) has dimension one in exterior degree three. As an ordinary algebra, its nondegenerate complex Clifford algebra is \(\mathbb C\oplus \mathbb C\); the underlying ordinary algebra factorization becomes \(M_2(\mathbb C)\oplus M_2(\mathbb C)\), in agreement with Remark [fcap-003Q]. The primitive relation, HKS argument, and factorization here are mathematical proofs without a packaged formal theorem.

                                                    Proposition. Normal ordering adds a matrix-trace central character [fcap-0035]

                                                      Let \(K\) be a field in which \(2\) is invertible. On \(V=M_N(K)\) take \[Q_{\mathrm {tr}}(Y)=\operatorname {Tr}(Y^2),\qquad B_{Q_{\mathrm {tr}}}(X,Y)=2\operatorname {Tr}(XY),\] and write \(d_{ij}=\iota _{Q_{\mathrm {tr}}}(E_{ij})\). The Clifford relation is \[d_{ij}d_{kl}+d_{kl}d_{ij} =2\delta _{jk}\delta _{li}.\] Let \(q(X)\) be the quadratic Clifford lift of the adjoint action \(Y\mapsto [X,Y]\). The normal-ordered lift used for the CAR module is \[D(X)=q(X)+\frac N2\operatorname {Tr}(X)1.\] On a matrix unit it has the concrete form \[D(E_{ij})=\frac 12\sum _{k=0}^{N-1}d_{ik}d_{kj}.\] Indeed, \(d_{ik}d_{kj}=\beta _{Q_{\mathrm {tr}}}(E_{ik},E_{kj}) +\delta _{ij}1\); summing the contraction term over \(k\) gives \((N/2)\delta _{ij}1\). This derives both the coefficient and its positive sign.

                                                      There are three different actions here. The ordinary exterior extension of the adjoint action is the derivation \[\delta _X(Y_1\wedge \cdots \wedge Y_r) =\sum _sY_1\wedge \cdots \wedge [X,Y_s]\wedge \cdots \wedge Y_r.\] The quadratic Clifford action is the inner derivation \([q(X),-]=[D(X),-]\). The CAR representation on the full Clifford algebra is instead left multiplication, \[\rho _{\mathrm {CAR}}(X)c=D(X)c, \qquad \rho _{\mathrm {CAR}}(X)c-cD(X)=[D(X),c].\] On the central matrix the three formulas read: \[\operatorname {ad}_I=0,\qquad q(I)=0, \qquad D(I)=\frac {N^2}{2}1.\] The left-regular carrier has dimension \(2^{N^2}\). For \(N\geq 1\) it is not an irreducible exterior spinor carrier; \(N=0\) is the one-dimensional trivial exception.

                                                      The exact identities are formalized at the pinned TauCeti revision. The normal-ordered lift, its matrix-unit and generator formulas, and the left-action comparison have the field and inverse-two hypotheses stated above. They do not identify this module with an irreducible spinor representation.

                                                      [pressley1986loop, Section 12.1, pp. 230--233] let the determinant-square-root cover of \(U(W)\) act on \(\bigwedge ^kW\) by \(u^{-1}\bigwedge ^k(A)\), with \(u^2=\det A\). Differentiating first along a unitary curve gives \(d\bigwedge (X)-\frac 12\operatorname {Tr}_W(X)\) for \(X\in \mathfrak u(W)\); complexifying then gives the corresponding \(\mathfrak {gl}(W)\) formula. No arbitrary complex matrix is being presented as the derivative of a unitary curve.

                                                      [panyushev2001exterior, Section 2, Example 2.5(3), preprint pp. 8--9] works over an algebraically closed field of characteristic zero and gives the ordinary spinor correction as \(\mathbb k_{-\nu }\otimes \bigwedge W\), where \(\nu \) is half the sum of the weights with multiplicity. Evaluating \(\nu \) is half the exterior trace. These two sources agree on the ordinary minus-half-trace normalization. They do not produce the positive \((N/2)\operatorname {Tr}\) term above: the trace of \(\operatorname {ad}_X\) on \(M_N\) is zero, while that term is the contraction created by normal ordering on the full left-regular Clifford module.

                                                      Proposition. The positive CAR product has half-staircase weight [fcap-0036]

                                                        Let \(K\) be a field in which \(2\) is invertible. Begin with \(N=2\) and put \(v_+=d_{01}\). The relations are \[d_{01}^2=d_{10}^2=0,\qquad d_{00}^2=d_{11}^2=1, \qquad d_{01}d_{10}+d_{10}d_{01}=2.\] The only raising operator is \[D(E_{01})=\frac 12(d_{00}d_{01}+d_{01}d_{11}),\] and each summand kills \(v_+\). For the diagonal operators, the same CAR relations give \[D(E_{00})v_+=\frac 32v_+,\qquad D(E_{11})v_+=\frac 12v_+.\] Thus \(v_+\) has weight \((3/2,1/2)\), and \(D(I)v_+=2v_+\). Subtracting the central correction \((N/2)\operatorname {Tr}\) leaves \((1/2,-1/2)\). In characteristic zero this is the usual \(\mathfrak {sl}_2\) value \(\rho \).

                                                        For general \(N\), order the indices as \(0<1<\cdots <N-1\) and form the lexicographically ordered product \[v_+=\prod _{0\leq i<j<N}^{\mathrm {lex}}d_{ij}.\] Every positive matrix unit is \(Q_{\mathrm {tr}}\)-null, and distinct positive matrix units are polar-orthogonal. Their Clifford generators therefore anticommute. PBW linear independence shows that the ordered product is nonzero over a field.

                                                        Let \(i<j\). In \[D(E_{ij})v_+=\frac 12\sum _kd_{ik}d_{kj}v_+,\] a term with \(k<j\) vanishes because the positive generator \(d_{kj}\) already occurs in \(v_+\). If \(k\geq j\), then \(i<k\); first anticommute \(d_{ik}\) past \(d_{kj}\), using \(i\ne j\), and the repeated positive factor \(d_{ik}\) makes the term zero. Hence every raising operator annihilates \(v_+\).

                                                        For a diagonal generator, \[D(E_{ii})v_+=\frac 12\sum _kd_{ik}d_{ki}v_+.\] The contributions for \(k<i\), \(k=i\), and \(k>i\) are respectively \(0\), \(v_+\), and \(2v_+\). Therefore \[D(E_{ii})v_+ =\frac 12\bigl (1+2|\{k:k>i\}|\bigr )v_+.\] For \(i\in \operatorname {Fin}(N)\), the weight and its quadratic-lift part are \[\lambda _i=N-i-\frac 12,\qquad \lambda _i-\frac N2=\frac {N-1}{2}-i,\qquad \sum _i\lambda _i=\frac {N^2}{2}.\] The last equality agrees with the direct central calculation \(D(I)=N^2/2\).

                                                        TauCeti defines this ordered product and proves its nonvanishing, raising equations, and diagonal weight. See the exact pinned product and PBW nonvanishing and half-staircase theorems. Nonvanishing needs a field; the highest-vector statement adds invertible \(2\). Identifying the subtracted weight with the rational staircase requires characteristic zero.

                                                        [panyushev2001exterior, Section 2, Example 2.5(1), preprint p. 8] identifies, over an algebraically closed field of characteristic zero, the reduced adjoint spin module with highest weight \(\rho \). The subtracted weight above is exactly that semisimple staircase. The added \((N/2)\operatorname {Tr}\) character records the reductive center of this normal-ordered left action. This comparison does not attribute the selected product or the full left-regular CAR module to Panyushev, and it makes no irreducibility or whole-module decomposition claim.

                                                        Backlinks

                                                          Theorem. The Killing-adjoint lift has coefficient one quarter [fcap-0020]
                                                          adjointCliffordHom_eq_sum_bivector
                                                          L∃∀N

                                                            Let \(K\) be a field in which \(2\) is invertible, and let \(L\) be a finite-dimensional \(K\)-Lie algebra whose Killing form \(\kappa \) is nondegenerate (the formal hypothesis is \(\operatorname {IsKilling}_K(L)\)). Set \(Q_\kappa (y)=\kappa (y,y)\). If \((b_i)\) is a basis and \((b_i^\vee )\) its Killing-dual basis, then for \(x\in L\) the quadratic Clifford lift of the adjoint action is \[\operatorname {adjointCliffordHom}(x) =\frac 14\sum _i\beta _{Q_\kappa } \bigl ([x,b_i],b_i^\vee \bigr ).\] Here \(\beta _{Q_\kappa }\) is the half-commutator bivector of Convention [fcap-000N]; the summand is the value of CliffordAlgebra.adjointBivector_apply.

                                                            Kostant identifies the complex orthogonal Lie algebra with \(\bigwedge ^2L\) and constructs the adjoint map from a complex semisimple \(L\) using its Killing pairing [kostant1997clifford, Sections 1.1--1.2, pp. 275--276]. This supports the complex adjoint construction. The coordinate expression and its \(1/4\) normalization are the calculation here; Tau Ceti's theorem uses the stated \(\operatorname {IsKilling}\) hypothesis over a wider field class.

                                                            Proof. 

                                                              The invariance of \(\kappa \) makes \(\operatorname {ad}_x\) skew-adjoint. Applying Theorem [fcap-001Z] to \(f=\operatorname {ad}_x\) first gives a factor \(1/2\) with the dual basis for \(B_{Q_\kappa }\). Since \(B_{Q_\kappa }=2\kappa \), that polar-dual basis is \(d_i=\tfrac 12 b_i^\vee \). Bilinearity of the bivector gives the second factor \(1/2\), and \((1/2)(1/2)=1/4\). The equality \(B_{Q_\kappa }=2\kappa \) is the convention encoded by TauCeti.LieAlgebra.polarBilin_killingQuadraticForm.

                                                              Proposition. The differential of the actual Spin action is the quadratic lift [fcap-0037]

                                                                Let \(V\) be a positive-dimensional finite-dimensional real vector space, and let \(Q\) be nondegenerate. Give \(\mathcal {C}\kern -2pt\ell (Q)\) its ordinary finite-dimensional topology. Inside \(\mathcal {C}\kern -2pt\ell (Q)^\times \), evenness, reverse norm one, and normalization of the embedded vector subspace are closed conditions and define a subgroup. This subgroup is the actual vector-generated \(\operatorname {Spin}(Q)\). Indeed, for any element \(x\) satisfying them, its twisted action lies in \(O(Q)\). Cartan--Dieudonne supplies a vector product \(u\) with the same twisted action. For \(z=u^{-1}x\), \[\alpha (z)\iota (v)=\iota (v)z\qquad (v\in V).\] Graded centrality makes \(z\) a nonzero scalar; no parity of \(u\) has been assumed in this step. If \(Q(w)\ne 0\), every nonzero scalar \(c\) is the two-vector product \[\iota (w)\,\iota \left (\frac {c}{Q(w)}w\right )=c1.\] Thus \(x\) is vector-generated, and its stated evenness and norm place it in Spin. The reverse inclusion is immediate from the carrier definition. The closed-subgroup theorem now gives this concrete Spin group its finite-dimensional real Lie-group structure.

                                                                For \(A\in \mathfrak {so}(V,B_Q)\), let \(b(A)\) be the normalized quadratic Clifford lift of Theorem [fcap-000R]. For every \(t\in \mathbb R\), \[g_A(t)=\exp \bigl (t b(A)\bigr )\in \operatorname {Spin}(Q), \qquad g_A(t)\,\iota (v)\,g_A(t)^{-1} =\iota \bigl (\exp (tA)v\bigr ).\] Indeed, reversal sends \(b(A)\) to \(-b(A)\), so the exponential is even and has reverse norm one. The commutator identity \([b(A),\iota (v)]=\iota (Av)\) gives the displayed conjugation formula by the exponential series. Hence \(g_A(t)\) satisfies the three closed conditions above and belongs to the actual Spin group.

                                                                The tangent Lie algebra at \(1\) is exactly the image of \(b\). One inclusion is supplied by the curves \(g_A\). Conversely, let \(X\) be tangent at \(1\). Differentiating evenness, reverse norm one, and vector normalization gives \[X\text { even},\qquad \operatorname {rev}(X)=-X, \qquad [X,\iota (V)]\subseteq \iota (V).\] Write \([X,\iota (v)]=\iota (Av)\). Differentiating preservation of \(Q\) makes \(A\) skew-adjoint. Then \(X-b(A)\) is even and commutes with every vector, so it is scalar; reverse antisymmetry forces that real scalar to vanish. Therefore \[T_1\operatorname {Spin}(Q)=b\bigl (\mathfrak {so}(V,B_Q)\bigr ), \qquad d_1(\operatorname {spinVectorAction})\bigl (b(A)\bigr )=A.\] This is the genuine differential statement whose algebraic commutator precursor appears in Theorem [fcap-000R] and whose polar-dual coordinate formula is Theorem [fcap-001Z].

                                                                If \(\pi :\mathcal {C}\kern -2pt\ell (Q)\to \operatorname {End}_{\mathbb R}(S)\) is a finite-dimensional real Clifford module, continuity of the power series gives \[\pi (g_A(t))=\exp \bigl (t\pi (b(A))\bigr ), \qquad d\pi _1\bigl (b(A)\bigr )=\pi (b(A)).\] A complex Clifford module may be regarded here as a real module; this does not assert a separately constructed complex Lie-group structure.

                                                                [meinrenken2013clifford, Proposition 2.12, p. 40; Proposition 3.2, pp. 53--54] proves the normalized bivector realization and the Spin exponential/action identity. [lawson2016spin, I.2, Remark 2.1 and (2.6)--(2.7), pp. 13--14] independently give the ambient Clifford exponential and its conjugation derivative. The carrier completion and tangent converse above identify the exact vector-generated group used here; Lawson's ambient exponential formula alone does not supply that membership.

                                                                For the convention match, Meinrenken's bilinear form \(B_M\) satisfies \(B_Q=2B_M\). If \((e_M^i)\) is \(B_M\)-dual to \((e_i)\), then the \(B_Q\)-dual vectors are \(e_M^i/2\), and \[b(A)=\frac 12\sum _i\beta _Q\left (Ae_i,\frac 12e_M^i\right ) =\frac 14\sum _i\beta _Q(Ae_i,e_M^i) =\frac 14\sum _i(Ae_i)e_M^i.\] The last equality uses \(\operatorname {tr}(A)=0\); it does not rescale the half-commutator \(\beta _Q\). TauCeti proves the algebraic lift and generator action at the pinned revision through CliffordAlgebra.quadraticLift_lie_ι and the coordinate formula CliffordAlgebra.soEquivQuadratic_eq_sum_bivector. The same normalization governs the actual compact differential below.

                                                                For the standard positive form \(Q_n^+\), TauCeti equips the closed range of Spin in Clifford units with a Lie-group structure. The continuous multiplicative equivalence TauCeti.CliffordAlgebra.realCliffordSpinContinuousMulEquivUnitsRange identifies this range with the original vector-generated carrier. Write \(\eta :\Lambda ^2\mathbb R^n\to \mathfrak {cl}^2(Q_n^+)\) for the exterior-bivector realization and \(\kappa :\Lambda ^2\mathbb R^n\to \mathfrak {so}(n)\) for the standard bivector Lie equivalence. In these coordinates the differential of the actual projection \(q:\operatorname {Spin}(n)\to SO(n)\) satisfies \[(d_1q)_{\mathrm {coord}}\bigl (\eta (z)\bigr )=\kappa (z), \qquad b(A)=\eta \bigl (\kappa ^{-1}(A)\bigr ).\] The first equality is TauCeti.CliffordAlgebra.realCliffordSpinToSpecialOrthogonalCoordinateLieHom_eq_bivectorEquivSo. Its proof differentiates conjugation by the Clifford exponential and identifies the resulting action with the commutator above. Thus neither an extra sign nor a factor of two is inserted in passing from the algebraic lift to the group differential.

                                                                There is also an all-signature exponential criterion. For the standard real form \(Q_{p,q}\), CliffordAlgebra.forall_expUnit_smul_mem_realCliffordSpinGroup_iff_mem_quadraticLieSubalgebra states, for an ambient Clifford element \(x\), \[\bigl (\forall t\in \mathbb R,\ \exp (tx)\in \operatorname {Spin}(Q_{p,q})\bigr ) \quad \Longleftrightarrow \quad x\in \mathfrak {cl}^2(Q_{p,q}).\] Here the exponential is first formed as a Clifford unit and membership is in the actual Spin carrier. This quantifies over every real \(t\); it does not identify quadratic elements from a single exponential value. The differential declaration uses the compact form \(Q_n^+\); the criterion ranges over all standard real signatures. The argument above continues to supply the assertion for an arbitrary nondegenerate \(Q\).

                                                                Related

                                                                  Lemma. A bivector acts by an infinitesimal rotation [meinrenken2013clifford, Section 2.2.10, Theorem 2.1, pp. 39--40] [fcap-000P]
                                                                  bivector_lie_ι
                                                                  L∃∀N

                                                                    For \(u,v,x\in M\), the commutator of the bivector with a Clifford generator is \[[\beta _Q(u,v),\iota (x)] =\iota \left (B_Q(v,x)u-B_Q(u,x)v\right ).\] The endomorphism \[A_{u\wedge v}(x)=B_Q(v,x)u-B_Q(u,x)v\] is skew-adjoint for \(B_Q\), since \[B_Q(A_{u\wedge v}x,y)+B_Q(x,A_{u\wedge v}y)=0.\] Thus commutation by a degree-two Clifford element preserves the generating module and realizes the elementary infinitesimal orthogonal transformation attached to \(u\wedge v\).

                                                                    Proof. 

                                                                      Expand the half-commutator and move \(\iota (x)\) past \(\iota (u)\) and \(\iota (v)\) using the Clifford relation. The cubic terms cancel; the two polar terms combine to the displayed generator. Symmetry of \(B_Q\) then makes the skew-adjointness equation cancel in pairs.

                                                                      Convention. Bivector normalizations [meinrenken2013clifford, Section 2.2.5, Proposition 2.5, p. 33] [fcap-000N]
                                                                      bivector_defbivector_lie_ι
                                                                      L∃∀N

                                                                        Meinrenken works throughout this chapter with finite-dimensional vector spaces over a characteristic-zero field. The cards below retain his normalization and formulas but follow TauCeti's extensions. Each card states the commutative-ring, module, invertible-\(2\), finite-dimensional, or nondegeneracy assumptions it needs.

                                                                        Meinrenken starts with a symmetric bilinear form \(B\) and the relation \[uv+vu=2B(u,v)1.\] For the associated quadratic form \(Q(v)=B(v,v)\), the polar form used here is \(B_Q=2B\).

                                                                        Meinrenken's quantization formula and the half-commutator agree: \[q(u\wedge v)=uv-B(u,v)1 =\frac 12(uv-vu)=:\beta (u,v).\] The induced action on a vector is \[[\beta (u,v),x] =B_Q(v,x)u-B_Q(u,x)v =2\bigl (B(v,x)u-B(u,x)v\bigr ).\] This is Meinrenken's formula \(-2\iota _{B(x,-)}(u\wedge v)\) and Kostant's formula \(-2\iota _x(u\wedge v)\) in the corresponding contraction convention [kostant1997clifford, Section 2.4, (12) and Theorem 8, pp. 283, 286]. The factor of two belongs to the passage from \(B\) to the unhalved polar form \(B_Q\); it is not an additional rescaling of \(\beta \).

                                                                        Theorem. Quadratic elements realize the orthogonal Lie algebra [meinrenken2013clifford, Section 2.2.10, Proposition 2.12, p. 40] [fcap-000R]
                                                                        soEquivQuadraticsoEquivQuadratic_lie_ι
                                                                        L∃∀N

                                                                          Meinrenken states this realization over a characteristic-zero field. TauCeti proves the extension below over any field of characteristic different from \(2\). Let \(K\) be such a field, let \(V\) be finite-dimensional, and let \(Q\) be nondegenerate. Write \(\mathfrak {so}(V,B_Q)\) for the Lie algebra of endomorphisms skew-adjoint for the polar form. Then commutation on generators gives a Lie equivalence \[\mathfrak {so}(V,B_Q)\simeq _{\mathrm {Lie}}\mathfrak {cl}^{(2)}(Q).\] If \(A\in \mathfrak {so}(V,B_Q)\) corresponds to \(z_A\in \mathfrak {cl}^{(2)}(Q)\), then \[[z_A,\iota (x)]=\iota (Ax)\qquad (x\in V).\] Combining this equivalence with Theorem [fcap-000Q] gives the basis-free chain \[\bigwedge _K^2V\simeq _{\mathrm {Lie}}\mathfrak {so}(V,B_Q) \simeq _{\mathrm {Lie}}\mathfrak {cl}^{(2)}(Q).\] Kostant identifies the same degree-two subspace with \(\operatorname {Lie}(\operatorname {Spin}(V))\) and the first equivalence with the differential of the double cover [kostant1997clifford, Section 2.4, Theorem 8, pp. 286--287].