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

Reference. Clifford algebras and lie theory [meinrenken2013clifford]

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

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

bivector_def bivector_lie_ι
L∃∀N

Definition. Clifford bivector [meinrenken2013clifford, Section 2.2.5, Proposition 2.5, p. 33] [fcap-000O]

bivectorExterior equivExterior_bivectorExterior
L∃∀N

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

Proof.

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

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

soEquivQuadratic soEquivQuadratic_lie_ι
L∃∀N

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

bivectorEquivSo_apply_ιMulti bivectorEquivSo_apply_ιMulti_mulVec
L∃∀N

Theorem. Reconstructing a skew Clifford lift in polar-dual coordinates [fcap-001Z]

soEquivQuadratic_eq_sum_bivector
L∃∀N

Proof.

Theorem. The Killing-adjoint lift has coefficient one quarter [fcap-0020]

adjointCliffordHom_eq_sum_bivector
L∃∀N

Proof.

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

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

cliffordInducedRep cliffordInducedRep_apply
L∃∀N

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

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

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

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

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

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

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

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

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

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

soEquivQuadratic soEquivQuadratic_lie_ι
L∃∀N

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

Proof.

Definition. Clifford bivector [meinrenken2013clifford, Section 2.2.5, Proposition 2.5, p. 33] [fcap-000O]

bivectorExterior equivExterior_bivectorExterior
L∃∀N