Contents

and

# Contents

## Statement

###### Proposition

Let $G$ be a finite dimensional simply connected compact Lie group, with Lie algebra denoted $\mathfrak{g}$

Then the Sullivan minimal model of the classifying space $B G$ is the graded algebra

$inv^\bullet(\mathfrak{g}) \; \coloneqq \; Sym \big( \mathfrak{g}^\ast[2] \big)^G$

of invariant polynomials on $\mathfrak{g}$, equipped with vanishing differential:

$\big( inv^\bullet(\mathfrak{g}), d = 0 \big) \overset{\simeq}{\longrightarrow} CE \big( \mathfrak{l} B G \big) \,.$
###### Proof

That the rational cohomology of $B G$ is given by $inv(\mathfrak{g})$

$inv^\bullet(\mathfrak{g}) \;\simeq\; H^\bullet \big( B G, k \big)$

is due to Chern 50, (11), Bott 73, p. 239 (5 of 15).

(Here $k$ is the ground field of characteristic zero).

That the rational cohomology algebra of $B G$, with trivial differential, is the minimal Sullivan model for $B G$ is discussed for instance in (Félix-Oprea-Tanré 08).

## References

Last revised on August 27, 2020 at 11:15:07. See the history of this page for a list of all contributions to it.