Reference. Clifford algebras and lie theory [meinrenken2013clifford]
✍️source

2013
Eckhard Meinrenken

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

Reference. Clifford algebras and lie theory [meinrenken2013clifford]
✍️source

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.

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

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

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. Creation, contraction, and the line operator [meinrenken2013clifford, Section 3.2.2, Proposition 3.5, p. 57] [fcap-001I]

wedge contract lineOperator
L∃∀N

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

bivectorExteriorEquivQuadraticLieSubalgebra bivectorLieEquiv
L∃∀N

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. PBW equivalence with the exterior algebra [meinrenken2013clifford, Proposition 2.6, pp. 33--34] [fcap-000J]

filtrationAssociatedGradedEquivExterior
L∃∀N

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

SpinPolarizationData
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

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

spinRep pinRep
L∃∀N

Theorem. The algebraic Pin extension [meinrenken2013clifford, Definition 3.2, pp. 51--52] [fcap-001F]

pinDoubleCoverOfSurjective pinDoubleCover
L∃∀N

Theorem. The algebraic Spin extension [meinrenken2013clifford, Definition 3.2, pp. 51--52] [fcap-001D]

spinDoubleCoverOfSurjective spinDoubleCover
L∃∀N

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

spinAction
L∃∀N

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

contract_wedge cliffordOperator_sq
L∃∀N

Example. The rank-one polarized model [meinrenken2013clifford, Section 3.2.4, Theorem 3.3, p. 59] [fcap-001P]

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

spinRep_apply pinRep_apply
L∃∀N