Lemma. A bivector acts by an infinitesimal rotation [meinrenken2013clifford, Section 2.2.10, Theorem 2.1, pp. 39--40] [fcap-000P]
✍️source
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.

      References

        Reference. Clifford algebras and lie theory [meinrenken2013clifford]

        • 2013
        • Eckhard Meinrenken
        @book{meinrenken2013clifford,
 title = {Clifford algebras and Lie theory},
 author = {Meinrenken, Eckhard},
 year = {2013},
 volume = {58},
 publisher = {Springer}
}

        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.

            § [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]
                ✍️source
                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.

                          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. They do not construct Lie-group structures or identify a differential of a Lie-group covering map. The next chapter instead reaches the global orthogonal group 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.

                              Backlinks

                                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.