invariant differential form

1 Lie Groups

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

Let $L_g:G⟶G$ and $R_g:G⟶G$ be the functions of left and right multiplication by $g$ (respectively). Let $C_g:G⟶G$ be the function of conjugation by $g$, i.e. $C_g(h):=gh{g}^{-1}$.

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

$$w_e(X_1,\mathrm{\dots },X_k),X_1,\mathrm{\dots },X_k\in T_e(G)$$

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

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

$$

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

where $\omega \in {\mathrm{\Omega }}^k(G)$ and $X_0,\mathrm{\dots },X_k$ are left invariant vector fields in $G$.

$$

Hence, the exterior derivative induces a map $d:{\mathrm{\Omega }}_L^k(G)⟶{\mathrm{\Omega }}_L^{k+1}(G)$ and $({\mathrm{\Omega }}_L^{*}(G),d)$ 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

$$(g,x)⟼t_g(x),g\in G,x\in M$$

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 $t_{(g,h)}(k):=gk{h}^{-1}$.

• 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 ${\mathrm{\Omega }}^k(M)$ the space of $k$-forms in $M$ and ${\mathrm{\Omega }}_G^k(M)$ 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 $J:{\mathrm{\Omega }}^k(M)⟶{\mathrm{\Omega }}_G^k(M)$ by

$$J(\omega )(X_1,\mathrm{\dots },X_k):=\frac{1}{\mu (G)}{\int }_G{t}_g^{*}\omega (X_1,\mathrm{\dots },X_k)𝑑\mu (g)$$

where $\omega \in {\mathrm{\Omega }}^k(M)$ and $X_1,\mathrm{\dots },X_k$ are vector fields of $M$.

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

$t_h(J(\omega ))(X_1,\mathrm{\dots },X_k)$ $=$ $J(\omega )({(t_h)}_{*}{X}_1,\mathrm{\dots },{(t_h)}_{*}{X}_k)$

$=$ ${\displaystyle \frac{1}{\mu (G)}}{\displaystyle {\int }_G}\omega ({(t_g)}_{*}{(t_h)}_{*}{X}_1,\mathrm{\dots },{(t_g)}_{*}{(t_h)}_{*}{X}_k)𝑑\mu (g)$

$=$ ${\displaystyle \frac{1}{\mu (G)}}{\displaystyle {\int }_G}\omega ({(t_{gh})}_{*}{X}_1,\mathrm{\dots },{(t_{gh})}_{*}{X}_k)𝑑\mu (g)$

$=$ ${\displaystyle \frac{1}{\mu (G)}}{\displaystyle {\int }_G}\omega ({(t_g)}_{*}{X}_1,\mathrm{\dots },{(t_g)}_{*}{X}_k)𝑑\mu (g)$

$=$ $J(\omega )(X_1,\mathrm{\dots },X_k)$

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

$J(\omega )(X_1,\mathrm{\dots },X_k)$ $=$ ${\displaystyle \frac{1}{\mu (G)}}{\displaystyle {\int }_G}{t}_g^{*}(\omega )(X_1,\mathrm{\dots },X_k)𝑑\mu (g)$

$=$ ${\displaystyle \frac{1}{\mu (G)}}{\displaystyle {\int }_G}\omega (X_1,\mathrm{\dots },X_k)𝑑\mu (g)$

$=$ $\omega (X_1,\mathrm{\dots },X_k)$

$$

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

$$

From the previous observations we can see that the exterior derivative takes invariant forms to invariant forms, inducing a map $d:{\mathrm{\Omega }}_G^k(M)⟶{\mathrm{\Omega }}_G^{k+1}(M)$. Hence, $({\mathrm{\Omega }}_G^{*}(M),d)$ 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 $H^k(M;ℝ)$. Hence, $G$ acts as a group of automorphisms on $H^k(M;ℝ)$. Let $H^k{(M;ℝ)}^G$ be the fixed point set of this action.

$$

Theorem - The inclusion $I:{\mathrm{\Omega }}_G^k(M)⟶{\mathrm{\Omega }}^k(M)$ induces an isomorphism

$$

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 $H^k(M;ℝ)$. Hence, the fixed point set is the whole $H^k(M;ℝ)$ and there is an isomorphism

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