View source for Weil algebra of a Lie algebra

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130050/w1300501.png" /> be a connected [[Lie group|Lie group]] with [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130050/w1300502.png" />. The Weil algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130050/w1300503.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130050/w1300504.png" /> was first introduced in a series of seminars by H. Cartan [[#References|[a1]]], in part based on some unpublished work of A. Weil. As a differential [[Graded algebra|graded algebra]], it is given by the tensor product

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130050/w1300505.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130050/w1300506.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130050/w1300507.png" /> denote the exterior and symmetric algebras, respectively (cf. also [[Exterior algebra|Exterior algebra]]; [[Symmetric algebra|Symmetric algebra]]).

The Weil algebra and its generalizations have been studied extensively by F.W. Kamber and Ph. Tondeur [[#References|[a3]]] [[#References|[a4]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130050/w1300508.png" /> be a maximal compact subgroup, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130050/w1300509.png" /> denoting the Lie algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130050/w13005010.png" />. The relative Weil algebra for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130050/w13005011.png" /> is defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130050/w13005012.png" /></td> </tr></table>

With regards to the universal classifying bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130050/w13005013.png" /> (cf. also [[Bundle|Bundle]]; [[Classifying space|Classifying space]]; [[Universal space|Universal space]]), there are canonical isomorphisms in [[Cohomology|cohomology]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130050/w13005014.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130050/w13005015.png" /> denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130050/w13005016.png" />-invariant polynomials. For a given integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130050/w13005017.png" />, one has the ideal

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130050/w13005018.png" /></td> </tr></table>

generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130050/w13005019.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130050/w13005020.png" />. This leads to the truncated Weil algebra

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130050/w13005021.png" /></td> </tr></table>

The cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130050/w13005022.png" /> plays a prominent role in the study of secondary characteristic classes (cf. also [[Characteristic class|Characteristic class]]) of foliations and foliated bundles [[#References|[a3]]] (see also [[#References|[a2]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Cartan,   "Cohomologie réelle d'un espace fibré principal differentiable" , ''Sem. H. Cartan 1949/50, Exp. 19–20''  (1950)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Dupont,   F.W. Kamber,   "On a generalization of Cheeger–Chern–Simons classes"  ''Illinois J. Math.'' , '''34'''  (1990)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F.W. Kamber,   Ph. Tondeur,   "Foliated bundles and characteristic classes" , ''Lecture Notes in Mathematics'' , '''493''' , Springer  (1975)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  F.W. Kamber,   Ph. Tondeur,   "Semi-simplicial Weil algebras and characteristic classes"  ''Tôhoku Math. J.'' , '''30'''  (1978)  pp. 373–422</TD></TR></table>