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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 was first introduced in a series of seminars by H. Cartan [[#References|[a1]]], in part based on some unpublished work of A. Weil.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>
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<!--This article has been texified automatically. Since there was no Nroff source code for this article,
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the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
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was used.
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If the TeX and formula formatting is correct and if all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
  
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]]).
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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
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Let $G$ be a connected [[Lie group|Lie group]] with [[Lie algebra|Lie algebra]] $\frak g$. The Weil algebra $W ( \mathfrak{g} )$ of $\frak g$ 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/w13005012.png" /></td> </tr></table>
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\begin{equation*} W ( \mathfrak { g } ) = \bigwedge \mathfrak { g } ^ { * } \bigotimes S \mathfrak { g } ^ { * }, \end{equation*}
  
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]]
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where $\wedge \mathfrak { g } ^ { * }$ and $S \mathfrak { g }  ^ { * }$ denote the exterior and symmetric algebras, respectively (cf. also [[Exterior algebra|Exterior algebra]]; [[Symmetric algebra|Symmetric 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/w13005014.png" /></td> </tr></table>
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The Weil algebra and its generalizations have been studied extensively by F.W. Kamber and Ph. Tondeur [[#References|[a3]]] [[#References|[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
  
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
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\begin{equation*} W ( G , K ) = \{ \bigwedge ( \mathfrak { g } / \mathfrak { k } ) ^ { * } \bigotimes S \mathfrak { g } ^ { * } \} ^ { K }. \end{equation*}
  
<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>
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With regards to the universal classifying bundle $E G \rightarrow B G$ (cf. also [[Bundle|Bundle]]; [[Classifying space|Classifying space]]; [[Universal space|Universal space]]), there are canonical isomorphisms in [[Cohomology|cohomology]]
  
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>
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$$
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\begin{array}{ccc}
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H^*(W(G,K)) & \xrightarrow[\cong]{\ \ w\ \ } & H^*(EG/K, \mathbf{R}) \\
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\cong \Big\downarrow& & \Big\downarrow \cong\\
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I(K) & \xrightarrow[\cong]{\ \ w\ \ } & H^*(BK, \mathbf{R})
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\end{array}
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$$
  
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]]]).
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where $I ( K )$ denotes the $\operatorname{Ad} K$-invariant polynomials. For a given integer $k \ge 0$, one has the ideal
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\begin{equation*} F W = F ^ { 2 ( k + 1 ) } W ( G , K ) \subseteq W ( G , K ), \end{equation*}
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generated by $S ^ { \text{l} } ( \mathfrak { g } ^ { * } )$, for $\text{l} \geq k + 1$. This leads to the truncated Weil algebra
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\begin{equation*} W _ { k } = W ( G , K ) _ { k } = W ( G , K ) / F W. \end{equation*}
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The cohomology $H ^ { * } ( W _ { k } )$ 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====
 
====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>
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<table>
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<tr><td valign="top">[a1]</td> <td valign="top"> H. Cartan, "Cohomologie réelle d'un espace fibré principal différentiable" , ''Sém. H. Cartan 1949/50, Exp. 19–20'' (1950)</td></tr>
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<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) {{MR|1046564}} {{ZBL|0724.57018}} </td></tr>
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<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) {{MR|0402773}} {{MR|0385886}} {{ZBL|0311.57011}} {{ZBL|0308.57011}} </td></tr>
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<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 {{MR|0509023}} {{ZBL|0398.57006}} </td></tr>
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</table>

Latest revision as of 18:56, 29 February 2024

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
How to Cite This Entry:
Weil algebra of a Lie algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weil_algebra_of_a_Lie_algebra&oldid=14594
This article was adapted from an original article by James F. Glazebrook (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article