# invariant differential form

## 1 Lie Groups

Let $G$ be a Lie group and $g\in G$.

Let $L_{g}:G\longrightarrow G$ and $R_{g}:G\longrightarrow G$ be the functions of left and right multiplication by $g$ (respectively). Let $C_{g}:G\longrightarrow G$ be the function of conjugation by $g$, i.e. $C_{g}(h):=ghg^{-1}$.

A differential $k$-form (http://planetmath.org/DifferentialForms) $\omega$ on $G$ is said to be

• left invariant$\,$ if $L_{g}^{*}\,\omega=\omega$ for every $g\in G$, where $L_{g}^{*}$ is the pullback induced $L_{g}$.

• right invariant$\,$ if $R_{g}^{*}\,\omega=\omega$ for every $g\in G$, where $R_{g}^{*}$ is the pullback induced $R_{g}$.

• invariant or bi-invariant$\,$ if it is both left invariant and right invariant.

• adjoint invariant$\,$ if $C_{g}^{*}\,\omega=\omega$ for every $g\in G$, where $C_{g}^{*}$ is the pullback induced by $C_{g}$.

$\,$

Much like left invariant vector fields (http://planetmath.org/LieGroup), left invariant forms are uniquely determined by their values in $T_{e}(G)$, the tangent space at the identity element $e\in G$, i.e. a left invairant form $\omega$ is uniquely determined by the values

 $w_{e}(X_{1},\dots,X_{k})\,,\qquad\qquad X_{1},\dots,X_{k}\in T_{e}(G)$

This means that left invariant forms are uniquely determined by their values on the Lie algebra of $G$.

Under this setting, the space $\Omega^{k}_{L}(G)$ of left invariant $k$-forms can be identified with $Hom(\Lambda^{k}\mathfrak{g},\mathbb{R})$, the space of homomorphisms from the $k$-th exterior power of $\mathfrak{g}$ to $\mathbb{R}$, where $\mathfrak{g}$ denotes the Lie algebra of $G$.

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- Let $\Omega^{k}(G)$ be the space of $k$-forms in $G$. The exterior derivative $d:\Omega^{k}(G)\longrightarrow\Omega^{k+1}(G)$ takes left invariant forms to left invariant forms. Moreover, the formula for exterior derivative for left invariant forms simplifies to

 $d\omega(X_{0},\dots,X_{k})=\sum_{i

where $\omega\in\Omega^{k}(G)$ and $X_{0},\dots,X_{k}$ are left invariant vector fields in $G$.

$\,$

Hence, the exterior derivative induces a map $d:\Omega^{k}_{L}(G)\longrightarrow\Omega^{k+1}_{L}(G)$ and $(\Omega^{*}_{L}(G),d)$ forms a chain complex. Thus, we can talk about the cohomology groups of left invariant forms.

Similar results hold for right invariant forms.

## 2 Manifolds

Suppose a Lie group $G$ acts smoothly (http://planetmath.org/Manifold) on a differential manifold $M$ and let

 $(g,x)\longmapsto t_{g}(x)\,,\qquad\qquad g\in G,x\in M$

denote the action of $G$.

A differential $k$-form $\omega$ in $M$ is said to be invariant if $t_{g}^{*}\,\omega=\omega$ for every $g\in G$, where $t_{g}^{*}$ denotes the pullback induced by $t_{g}$.

This definition reduces to the previous ones when we take $M$ as the group $G$ itself and when the action is

• the action of $G$ on itself by left multiplication.

• the action of $G$ on itself by right multiplication.

• the action of $G\times G$ on $G$ defined by $t_{(g,h)}(k):=gkh^{-1}$.

• the action of $G$ on itself by conjugation.

## 3 Compact Lie Group Actions

We now consider actions of a compact Lie group $G$ on a manifold $M$. Let $\Omega^{k}(M)$ the space of $k$-forms in $M$ and $\Omega_{G}^{k}(M)$ the space of invariant $k$-forms in $M$. Let $\mu$ be the Haar measure of $G$.

From each $k$-form in $M$ we can construct an invariant form by taking on its ””. Following this idea we define a map $J:\Omega^{k}(M)\longrightarrow\Omega_{G}^{k}(M)$ by

 $J(\omega)\,(X_{1},\dots,X_{k}):=\frac{1}{\mu(G)}\int_{G}t_{g}^{*}\,\omega\,(X_% {1},\dots,X_{k})\;d\mu(g)$

where $\omega\in\Omega^{k}(M)$ and $X_{1},\dots,X_{k}$ are vector fields of $M$.

The image of the map $J$ is indeed in $\Omega_{G}^{k}(M)$ since for every $h\in G$:

 $\displaystyle t_{h}(J(\omega))\,(X_{1},\dots,X_{k})$ $\displaystyle=$ $\displaystyle J(\omega)\,((t_{h})_{*}X_{1},\dots,(t_{h})_{*}X_{k})$ $\displaystyle=$ $\displaystyle\frac{1}{\mu(G)}\int_{G}\omega((t_{g})_{*}(t_{h})_{*}X_{1},\dots,% (t_{g})_{*}(t_{h})_{*}X_{k})\;d\mu(g)$ $\displaystyle=$ $\displaystyle\frac{1}{\mu(G)}\int_{G}\omega((t_{gh})_{*}X_{1},\dots,(t_{gh})_{% *}X_{k})\;d\mu(g)$ $\displaystyle=$ $\displaystyle\frac{1}{\mu(G)}\int_{G}\omega((t_{g})_{*}X_{1},\dots,(t_{g})_{*}% X_{k})\;d\mu(g)$ $\displaystyle=$ $\displaystyle J(\omega)\,(X_{1},\dots,X_{k})$

Moreover, $J$ is the identity for invariant $k$-forms. Suppose $\omega\in\Omega_{G}^{k}(M)$, then

 $\displaystyle J(\omega)\,(X_{1},\dots,X_{k})$ $\displaystyle=$ $\displaystyle\frac{1}{\mu(G)}\int_{G}t_{g}^{*}(\omega)\,(X_{1},\dots,X_{k})\;d% \mu(g)$ $\displaystyle=$ $\displaystyle\frac{1}{\mu(G)}\int_{G}\omega\,(X_{1},\dots,X_{k})\;d\mu(g)$ $\displaystyle=$ $\displaystyle\omega(X_{1},\dots,X_{k})$

$\,$

The map $J$ is a chain map, i.e. $dJ=Jd$, where $d$ is the exterior derivative of a form.

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From the previous observations we can see that the exterior derivative takes invariant forms to invariant forms, inducing a map $d:\Omega_{G}^{k}(M)\longrightarrow\Omega_{G}^{k+1}(M)$. Hence, $(\Omega_{G}^{*}(M),d)$ is a chain complex and we can talk about the cohomology groups of invariant forms in $M$.

## 4 Cohomology of Manifolds

Let $G$ be a compact Lie group that acts smoothly on a manifold $M$ (again, with the action denoted by $t_{g}$).

Since $t_{g}$ is a diffeomorphism of $M$ it induces an automorphism $t_{g}^{*}$ on the cohomology groups $H^{k}(M;\mathbb{R})$. Hence, $G$ acts as a group of automorphisms on $H^{k}(M;\mathbb{R})$. Let $H^{k}(M;\mathbb{R})^{G}$ be the fixed point set of this action.

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Theorem - The inclusion $I:\Omega_{G}^{k}(M)\longrightarrow\Omega^{k}(M)$ induces an isomorphism

 $\xymatrix{I^{*}:H^{k}(\Omega_{G}(M))\ar[r]^{\simeq}&H^{k}(M;\mathbb{R})^{G}}$

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If in $G$ is connected, then $t_{g}$ and the identity $1_{M}$ are homotopic, $t_{g}\simeq 1_{M}$, for every $g\in G$. This implies that the induced automorphisms are the same, i.e. $t_{g}^{*}=Id$, where $Id$ is the identity on $H^{k}(M;\mathbb{R})$. Hence, the fixed point set is the whole $H^{k}(M;\mathbb{R})$ and there is an isomorphism

 $\xymatrix{I^{*}:H^{k}(\Omega_{G}(M))\ar[r]^{\simeq}&H^{k}(M;\mathbb{R})}$

Thus, the cohomology groups of a manifold where a compact connected Lie group acts are just the cohomology groups defined by the invariant forms on $M$. This means we can ”forget” the whole of differential forms in $M$ and regard only those who are invariant.

 Title invariant differential form Canonical name InvariantDifferentialForm Date of creation 2013-03-22 17:48:31 Last modified on 2013-03-22 17:48:31 Owner asteroid (17536) Last modified by asteroid (17536) Numerical id 25 Author asteroid (17536) Entry type Definition Classification msc 58A10 Classification msc 57T10 Classification msc 57S15 Classification msc 22E30 Classification msc 22E15 Synonym invariant form Synonym bi-invariant form Synonym bi-invariant differential form Related topic CohomologyOfCompactConnectedLieGroups Defines left invariant differential form Defines left invariant form Defines right invariant differential form Defines right invariant form Defines adjoint invariant form Defines adjoint invariant differential form Defines chain complex of invariant forms Defines cohomology of manifolds with a Lie group action