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

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.

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

Reference. Clifford algebras and lie theory [meinrenken2013clifford]

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]

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]
✍️source

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

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

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

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

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