nLab
invariant polynomial

Idea

For every Lie algebra or ∞-Lie algebra or ∞-Lie algebroid $𝔞$ there is its Chevalley-Eilenberg algebra $\mathrm{CE}(𝔞)$ and its Weil algebra $W(𝔞)$ and a dg-algebra morphism

$$\mathrm{CE}(𝔞)\leftarrow W(𝔞).$$CE(\mathfrak{a}) \leftarrow W(\mathfrak{a}) \,.

The dg-algebra $\mathrm{inv}(𝔞)$ of invariant polynomials on $𝔞$ sits in the kernel of this map

$$\mathrm{CE}(𝔞)\leftarrow W(𝔞)\leftarrow \mathrm{inv}(𝔞).$$CE(\mathfrak{a}) \leftarrow W(\mathfrak{a}) \leftarrow inv(\mathfrak{a}) \,.

Definition for ordinary Lie algebras

For $𝔤$ a Lie algebra with Chevalley-Eilenberg algebra $\mathrm{CE}(𝔤)$ and Weil algebra $W(𝔤)$, an invariant polynomial on $𝔤$ is an elements $P\in W(𝔞)$ with the property that

• $P$ is a wedge product of generators in the shifted copy of $C^{\mathrm{\infty }}(𝔤)$ in $W(𝔤)$;

• it is closed in $W(𝔤)$ in that $d_{W(𝔤)}P=0$

Definition for $\mathrm{\infty }$-Lie algebroids

For $𝔞$ a Lie infinity-algebroid with Chevalley-Eilenberg algebra $\mathrm{CE}(𝔞)$ and Weil algebra $W(𝔞)$, the dg-algebra of invariant polynomials on $𝔞$ is the sub-dg-algebra of $W(𝔞)$ generated by those elements $P\in W(𝔞)$ with the property that

• $P$ is a wedge product of generators in the shifted copy of $C^{\mathrm{\infty }}(𝔞)$ in $W(𝔞)$;

• it is closed in $W(𝔤)$ in that $d_{W(𝔤)}P=0$

Urs Schreiber: there is a longer story to be told here, will come back to that…