Weil algebra of a Lie algebra

Let $G$ be a connected Lie group with Lie algebra $\frak g$. The Weil algebra $W ( \mathfrak{g} )$ of $\frak g$ was first introduced in a series of seminars by H. Cartan [a1], in part based on some unpublished work of A. Weil. As a differential graded algebra, it is given by the tensor product

\begin{equation*} W ( \mathfrak { g } ) = \bigwedge \mathfrak { g } ^ { * } \bigotimes S \mathfrak { g } ^ { * }, \end{equation*}

where $\wedge \mathfrak { g } ^ { * }$ and $S \mathfrak { g } ^ { * }$ denote the exterior and symmetric algebras, respectively (cf. also Exterior algebra; Symmetric algebra).

The Weil algebra and its generalizations have been studied extensively by F.W. Kamber and Ph. Tondeur [a3] [a4]. Let $K \subseteq G$ be a maximal compact subgroup, with $\frak p$ denoting the Lie algebra of $K$. The relative Weil algebra for $( G , K )$ is defined by

\begin{equation*} W ( G , K ) = \{ \bigwedge ( \mathfrak { g } / \mathfrak { k } ) ^ { * } \bigotimes S \mathfrak { g } ^ { * } \} ^ { K }. \end{equation*}

With regards to the universal classifying bundle $E G \rightarrow B G$ (cf. also Bundle; Classifying space; Universal space), there are canonical isomorphisms in cohomology

$$ \begin{array}{ccc} H^*(W(G,K)) & \xrightarrow[\cong]{\ \ w\ \ } & H^*(EG/K, \mathbf{R}) \\ \cong \Big\downarrow& & \Big\downarrow \cong\\ I(K) & \xrightarrow[\cong]{\ \ w\ \ } & H^*(BK, \mathbf{R}) \end{array} $$

where $I ( K )$ denotes the $\operatorname{Ad} K$-invariant polynomials. For a given integer $k \ge 0$, one has the ideal

\begin{equation*} F W = F ^ { 2 ( k + 1 ) } W ( G , K ) \subseteq W ( G , K ), \end{equation*}

generated by $S ^ { \text{l} } ( \mathfrak { g } ^ { * } )$, for $\text{l} \geq k + 1$. This leads to the truncated Weil algebra

\begin{equation*} W _ { k } = W ( G , K ) _ { k } = W ( G , K ) / F W. \end{equation*}

The cohomology $H ^ { * } ( W _ { k } )$ plays a prominent role in the study of secondary characteristic classes (cf. also Characteristic class) of foliations and foliated bundles [a3] (see also [a2]).

References

[a1]	H. Cartan, "Cohomologie réelle d'un espace fibré principal différentiable" , Sém. H. Cartan 1949/50, Exp. 19–20 (1950)
[a2]	J.L. Dupont, F.W. Kamber, "On a generalization of Cheeger–Chern–Simons classes" Illinois J. Math. , 34 (1990) MR1046564 Zbl 0724.57018
[a3]	F.W. Kamber, Ph. Tondeur, "Foliated bundles and characteristic classes" , Lecture Notes in Mathematics , 493 , Springer (1975) MR0402773 MR0385886 Zbl 0311.57011 Zbl 0308.57011
[a4]	F.W. Kamber, Ph. Tondeur, "Semi-simplicial Weil algebras and characteristic classes" Tôhoku Math. J. , 30 (1978) pp. 373–422 MR0509023 Zbl 0398.57006