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

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.

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

Reference. The exterior algebra and Spin of an orthogonal g-module [panyushev2001exterior]

Reference. Clifford algebra analogue of the Hopf--Koszul--Samelson theorem [kostant1997clifford]

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]

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]
✍️source

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]

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

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

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

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

cliffordInducedRep cliffordInducedRep_apply
L∃∀N

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

adjointCliffordHom_eq_sum_bivector
L∃∀N

Proof.