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

✍️sourceAGENTDRAFTED

Let \(\mathfrak g\) be finite-dimensional complex semisimple of rank \(\ell \). Write \(I=(\bigwedge \mathfrak g)^{\mathfrak g}\) for the adjoint invariants and \(I^+\) for their positive-degree part. Extend the Killing form \(\kappa \) to the determinant pairing \(\langle -,-\rangle \) on each exterior degree, with different degrees orthogonal. Let \(\pi :\bigwedge \mathfrak g\to I\) be the invariant projection along \(\mathfrak g\mathbin {\cdot }\bigwedge \mathfrak g\), the span of all adjoint derivatives. The coproduct on \(I\) is \[\Delta =(\pi \otimes \pi )\Delta _0|_I,\] where \(\Delta _0(x)=x\otimes 1+1\otimes x\) for \(x\in \mathfrak g\). It is this projected coproduct that defines primitive invariants: \[P=\{p\in I:\Delta p=p\otimes 1+1\otimes p\} =I^+\cap ((I^+)^2)^\perp .\] The ordinary exterior coproduct need not take \(I\) into \(I\otimes I\). The Lie-homology construction makes \(I\), with the displayed \(\Delta \), a graded super Hopf algebra; multiplication on \(I\otimes I\) uses the Koszul rule \((a\otimes b)(c\otimes d)=(-1)^{\lvert b\rvert \lvert c\rvert }ac\otimes bd\) for homogeneous elements. The coproduct is dual to the wedge product. This is the input from [meinrenken2013clifford, Theorem 10.1 and Proposition 10.11, pp. 240--241].

The Hopf--Koszul--Samelson theorem says that inclusion extends to \[\bigwedge P\xrightarrow {\sim }I.\] Here every primitive has odd exterior degree. Indeed, a nonzero homogeneous even primitive \(p\) would have a least nilpotent power \(p^k=0\); the component \(k p\otimes p^{k-1}\) of \(\Delta (p^k)\) could not vanish in characteristic zero. The orthogonal decomposition \(I^+=P\oplus (I^+)^2\) then proves generation: substituting it repeatedly expresses every element as a polynomial in \(P\), since the augmentation ideal is nilpotent. For independence, contraction by a primitive acts as an odd derivation on \(I\), and contraction of two primitives is a scalar. The first assertion is dual to \(\Delta \) preserving multiplication; the second follows from \(P\perp (I^+)^2\). Successive contractions by a dual primitive basis extract the coefficients of an exterior-monomial relation, starting with the largest number of factors. All coefficients vanish. This proves the stated isomorphism; compare [meinrenken2013clifford, Theorem 10.2 and Proposition 10.13, pp. 242--243]. By Theorem [fcap-003P], \(\dim I=2^\ell \), so \(\dim P=\ell \).

For \(x\in \mathfrak g\), let \(\iota _x\) be contraction using \(\kappa \), and set \(\iota _{x_1\wedge \cdots \wedge x_k}=\iota _{x_1}\cdots \iota _{x_k}\). Exterior reversion is \(\operatorname {rev}(u)=(-1)^{k(k-1)/2}u\) on degree \(k\). The form relevant to Clifford multiplication is \[B_\wedge (u,v)=(\iota _u v)_0 =\langle \operatorname {rev}(u),v\rangle ,\qquad Q_P(p)=B_\wedge (p,p).\] In particular, on primitive degree \(2m+1\), \(B_\wedge (p,q)=(-1)^m\langle p,q\rangle \). The Clifford polar relation for \(Q_P\) is therefore \(pq+qp=2B_\wedge (p,q)\), with the same generator-square convention as \(Q_\kappa (x)=\kappa (x,x)\).

This restriction of \(B_\wedge \) to \(P\) is nondegenerate. Choose a compact real form. Its Killing form is negative definite; on exterior degree \(k\), the determinant pairing is \((-1)^k\) times a positive definite real form. The subspace \(P\) is graded and preserved by compact conjugation, so the determinant pairing and its degreewise sign multiple \(B_\wedge \) are nondegenerate there. This also explains the normalization change from [meinrenken2013clifford, Section 11.3, p. 260], which starts with a positive compact form. Replacing that form by its negative preserves the graded orthogonal subspaces while changing the indicated degree signs.