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created invariant polynomial
yesterday or so I started, finally, expanding and polishing invariant polynomial. But the main punchline is still missing, I’ll include it later:
I think the good treatment works at the level of cosimplicial algebras, not dg-algebras. Let $\mathfrak{g}$ be the Lie algebra of a Lie group regarded as an infinitesimal oo-Lie groupoid, i.e. as the simplicial object
$\mathfrak{g} = \left( \cdots (G \times G)_{(1)} \stackrel{\to}{\to} G_{(1)} * \right)$where the subscript denotes first order infinitesimal neighbourhood of the neutral element. This is such that the cosimplicial algebra of functions on this maps under forming normalized cochains to the Chevalley-Eilenberg algebra of $\mathfrak{g}$.
Then we can apply the infinitesimal path oo-groupoid functor to get the corresponding tangent Lie oo-algebroid $T \mathfrak{g}$. This is such that the cosimplicial algebra of functions on this maps under forming normalized chains to the Weil algebra of $\mathfrak{g}$.
Then we can form the cokernel $\mathfrak{g} \to T \mathfrak{g} \to T \mathfrak{g} / \mathfrak{g}$.
And that gives the invariant polynomials: the normalized chains dg-algeba of functions on $T\mathfrak{g}/\mathfrak{g}$ is the complex whose cohomology are the invariant polyomials on $\mathfrak{g}$.
In the relevant model structure the diagram $T \mathfrak{g} \leftarrow \mathfrak{g} \to *$ is cofibrant, so that this is indeed the homotopy cofiber.
So this solves the question of how to abstractly characterize the invariant polynomials as those element in the Weil algebra $W(\mathfrak{g})$ that are closed and in addition are not only are annihilated by projecting onto the Chevalley-Eilenberg algebra, but which have the property that they are entirely in the image generated by the shifted generators of the Weil algebra.
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