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Coadjoint Orbits

Coadjoint orbits are examples of symplectic manifolds. This post aims at defining coadjoint orbits with a review on Lie group action.

Author

Affiliation

Fei Ye

QCC-CUNY

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In this post, we will denote by $G$ a Lie group. A very good exposition on Lie group action can be found in Section 9.3 in (Marsden and Ratiu 1999).

1 Lie Group Actions

Let $X$ be a smooth manifold. A Lie group action of $G$ on $X$ is defined to be a smooth group morphism $\mathrm{\Psi }:G\to \mathrm{Diff}(X)$ in the sense that $\psi :G\times X\to X$, $\psi (g,x)={\mathrm{\Psi }}_g(x)$ is a smooth map, where $\mathrm{Diff}(X)$ is the group of diffeomorphisms of $X$ and ${\mathrm{\Psi }}_g:=\mathrm{\Psi }(g)$.

Generally, a group action on a space is a group homomorphism from a given group to the group of transformations of the space.

For a point $x$ in $X$, consider the evaluation map: $$\begin{array}{rl}{o}_x:G& \to X\\ g& \mapsto g\cdot x:={\mathrm{\Psi }}_g(x).\end{array}$$ This map $o_x$ is known as the orbit map which is smooth by the definition of smooth action.

The $G$-orbit (or simply the orbit) of $x$ in $X$ is the image of $o_x$: $${\mathrm{Orb}}_x=o_x(G)=\{g\cdot x|g\in G\},$$ where $g\cdot x:={\mathrm{\Psi }}_g(x)$.

The stabilizer of $x$ in $X$ is the preimage of $o$ $$G_x=o_x^{-1}(x)=\{g|g\cdot x=x\}\subset G.$$

Since the action $\mathrm{\Psi }$ is smooth, the stabilizer $G_x=o_x^{-1}(x)$ is a closed submanifold of $G$. Moreover, $G_x$ is a subgroup of $G$. Hence $G_x$ is a Lie subgroup by Cartan’s Theorem.

Note by $L_g:G\to G$, $h\mapsto gh$ the left multiplication. Then $$\begin{array}{rl}(o_x\circ L_g)(h)=& o_x(gh)={\mathrm{\Psi }}_{gh}(x)\\ =& ({\mathrm{\Psi }}_g\circ {\mathrm{\Psi }}_h)(x)={\mathrm{\Psi }}_g({\mathrm{\Psi }}_h(x))\\ =& {\mathrm{\Psi }}_g(o_x(h))=({\mathrm{\Psi }}_g\circ o_x)(h).\end{array}$$

Therefore, $(\mathrm{d}{o}_x{)}_{gh}\circ (\mathrm{d}{L}_g{)}_h=(\mathrm{d}{\mathrm{\Psi }}_g{)}_{h\cdot x}\circ (\mathrm{d}{o}_x{)}_h$ for any $g,h$ in $G$. Since $L_g$ and ${\mathrm{\Psi }}_g$ are both diffeomorphisms, the equality implies that $\mathrm{d}{o}_x$ is a function on $G$ of constant rank. By the constant rank theorem, the orbit ${\mathrm{Orb}}_x$ is an immersed submanifold. The manifold structure on ${\mathrm{Orb}}_x$ is induced by the orbit map $o_x$. In general, the orbit ${\mathrm{Orb}}_x$ is not a submanifold of $X$, in the subset topology.

However, if $G$ is compact, then ${\mathrm{Orb}}_x$ is a submanifold of $X$ (Theorem 2.3 in (Gorbatsevich, Onishchik, and Vinberg 1993)).

Denote by $G/G_x$ the set of cosets. Then $G/G_x$ is a smooth manifold. The smooth map $G\to G/G_x$ is a locally trivial fiber bundle (Theorem I.4.8. in (Audin 2012)). Consequently, the tangent space of $T_e(G/G_x)$ can be identified with $\mathfrak{g}/{\mathfrak{g}}_x=T_e(G)/T_e(G_x)$.

Induced by the orbit map $o_x$, we get a bijection $$\begin{array}{rl}G/G_x& \to {\mathrm{Orb}}_x\\ g{G}_x& \mapsto g\cdot x.\end{array}$$ The $G$-action on $X$ induces a $G$-action on ${\mathrm{Orb}}_x$. Moreover, the $G$-action on ${\mathrm{Orb}}_x$ is transitive with the stabilizer $G_x$. Hence $G/G_x$ is diffeomorphic to ${\mathrm{Orb}}_x$ (Theorem 3.3 in (Gorbatsevich, Onishchik, and Vinberg 1993)).

1.1 Conjugation Action

Consider the map $\mathrm{\Psi }:G\to \mathrm{Aut}(G)$ sending an element $g$ to the inner automorphism ${\mathrm{\Psi }}_g$, i.e. ${\mathrm{\Psi }}_g(h)=gh{g}^{-1}$. Then $\mathrm{\Psi }$ is a group homomorphism. Indeed, for any $x,g,h\in G$, we have $$\mathrm{\Psi }(gh)(x)=(gh)x(gh{)}^{-1}=g(hx{h}^{-1})g^{-1}={\mathrm{\Psi }}_g{\mathrm{\Psi }}_h(x).$$ Therefore, $\mathrm{\Psi }(gh)={\mathrm{\Psi }}_g{\mathrm{\Psi }}_h$.

Moreover, since $G$ is a Lie group, the morphism $\mathrm{\Psi }$ is smooth. Indeed, $\mathrm{\Psi }$ can be viewed as the composition of multiplication and inverse operations: $$\begin{array}{ccccccc}G\times G& \to & G\times G& \to & G\times G& \to & G\\ (g,x)& \mapsto & (gx,g)& \mapsto & (gx,g^{-1})& \mapsto & gx{g}^{-1}\end{array}$$

This action is known as the conjugation action.

2 Representations of a Lie Group

A representation of a Lie group $G$ is a continuous group action on a vector space $V$: a group homomorphism $$\mathrm{\Pi }:G\to \mathrm{GL}(V)$$ that is continuous in the sense that $G\times V\to V$, $(g,v)\mapsto \mathrm{\Pi }(g)(v)$ is continuous.

2.1 Adjoint Representation of a Lie Group

Let $G$ be a Lie group and $e$ be the identity element of $G$. The tangent space $T_{e}G$ can be identified with the vector space of left invariant vector fields on $G$ and admits a Lie algebra structure defined by $[X,Y]=XY-YX$. This Lie algebra, denoted as $\mathfrak{g}$, is called the Lie algebra associated to $G$.

For each $g$ in $G$, we define ${\mathrm{Ad}}_g=(\mathrm{d}{\mathrm{\Psi }}_g{)}_e:\mathfrak{g}\to \mathfrak{g}$ to be the derivative of the inner automorphism ${\mathrm{\Psi }}_g:G\to G$ at the origin. Let $\exp (tX)$ be the unique integral curve associated to $X$. Then $$\begin{array}{rl}{\mathrm{Ad}}_g(X)=& (\mathrm{d}{\mathrm{\Psi }}_g{)}_e(X)\\ =& \frac{\mathrm{d}}{\mathrm{d}t}{|}_{t=0}({\mathrm{\Psi }}_g(\exp (tX)))\\ =& \frac{\mathrm{d}}{\mathrm{d}t}{|}_{t=0}(g\exp (tX)g^{-1}).\end{array}$$

As the exponential map $\exp :\mathfrak{g}\to G$ commutes with Lie group homomorphism and its derivative, we see that $$\exp ({\mathrm{Ad}}_g(X))=g\exp (X)g^{-1}$$

Because the pushforward of a diffeomorphism commutes with Lie brackets (see, for example, Corollary 8.31 in (Lee 2012)), and inner automorphisms and left multiplications of Lie groups are all diffeomorphisms. Then $${\mathrm{Ad}}_g([X,Y])=[{\mathrm{Ad}}_g(X),{\mathrm{Ad}}_g(Y)]$$ which means that ${\mathrm{Ad}}_g$ is a Lie algebra automorphism of $\mathfrak{g}$.

Consider the map $$\begin{array}{rl}\mathrm{Ad}:G& \to \mathrm{Aut}(\mathfrak{g})\\ g& \mapsto {\mathrm{Ad}}_g.\end{array}$$ For any $g$ and $h$ in $G$, we have $$\begin{array}{rl}\mathrm{Ad}(gh)=& {\mathrm{Ad}}_{gh}=(\mathrm{d}{\mathrm{\Psi }}_{gh}{)}_e\\ =& (\mathrm{d}({\mathrm{\Psi }}_g\circ {\mathrm{\Psi }}_h){)}_e=(\mathrm{d}{\mathrm{\Psi }}_g{)}_{{\mathrm{\Psi }}_h(e)}\circ (\mathrm{d}{\mathrm{\Psi }}_h{)}_e\\ =& (\mathrm{d}{\mathrm{\Psi }}_g{)}_e\circ (\mathrm{d}{\mathrm{\Psi }}_h{)}_e=\mathrm{Ad}(g)\mathrm{Ad}(h).\end{array}$$ Hence, $\mathrm{Ad}$ is a group representation called the adjoint representation of $G$.

2.2 Coadjoint Representation of a Lie Group

Let $G$ be a Lie group, $\mathfrak{g}$ the associated Lie algebra and ${\mathfrak{g}}^{\ast }$ the dual vector space of the Lie algebra $\mathfrak{g}$. The coadjoint representation of $G$, ${\mathrm{Ad}}^{\ast }:G\to \mathrm{Aut}({\mathfrak{g}}^{\ast })$, is defined by the identity $$<{\mathrm{Ad}}_g^{\ast }\mu ,Y>=<\mu ,{\mathrm{Ad}}_{g^{-1}}Y>$$ for all $g\in G$, $Y\in \mathfrak{g}$, and $\mu \in {\mathfrak{g}}^{\ast }$.

3 Infinitesimal Actions

Differentiating a group action at the origin induces an infinitesimal action on Lie algebras. In particular, the infinitesimal actions of adjoint and coadjoint representation of a connected Lie group $G$ are nothing but the adjoint and coadjoint representation of the Lie algebra $\mathfrak{g}$.

3.1 Adjoint Representation of a Lie Algebra

Consider the differential $(\mathrm{d}\mathrm{Ad}{)}_e$ of the adjoint representation of $G$. For any $X$ and $Y$ in $\mathfrak{g}$, we have $$\begin{array}{rl}& (\mathrm{d}\mathrm{Ad}{)}_e(X)(Y)\\ =& \frac{\mathrm{d}}{\mathrm{d}s}{|}_{s=0}({\mathrm{Ad}}_{\exp (sX)}(Y))\\ =& \frac{\mathrm{d}}{\mathrm{d}s}{|}_{s=0}(\frac{\mathrm{d}}{\mathrm{d}t}{|}_{t=0}\exp (sX)\exp (tY)\exp (-sX))\\ =& \frac{\mathrm{d}}{\mathrm{d}s}{|}_{s=0}(\exp (sX)Y\exp (-sX))\\ =& (\exp (sX)XY\exp (-sX)-\exp (sX)Y\exp (-sX))X{|}_{s=0}\\ =& [X,Y].\end{array}$$

We denote $(\mathrm{d}\mathrm{Ad}{)}_e$ as $\mathrm{ad}$ and $\mathrm{ad}(X)$ as ${\mathrm{ad}}_X$. It can be checked that $\mathrm{ad}([X,Y])=[\mathrm{ad}(X),\mathrm{ad}(Y)]$. Indeed, for any $Z$, we have $$\begin{array}{rl}\mathrm{ad}([X,Y])(Z)=& [[X,Y],Z]\\ =& [X,[Y,Z]]-[Y,[X,Z]]\\ =& {\mathrm{ad}}_X{\mathrm{ad}}_Y(Z)-{\mathrm{ad}}_Y{\mathrm{ad}}_X(Z).\end{array}$$ Therefore, $\mathrm{ad}:\mathfrak{g}\to \mathfrak{gl}(\mathfrak{g})$ defines a Lie algebra representation which is called the adjoint representation of the Lie algebra $\mathfrak{g}$.

3.2 Coadjoint Representation of a Lie Algebra

From the definition of ${\mathrm{Ad}}^{\ast }$, one can check that the differential ${\mathrm{ad}}^{\ast }=(\mathrm{d}{\mathrm{Ad}}^{\ast }{)}_e:\mathfrak{g}\to \mathfrak{gl}({\mathfrak{g}}^{\ast })$ is a representation of $\mathfrak{g}$ in ${\mathfrak{g}}^{\ast }$, which is called the coadjoint representation of ${\mathfrak{g}}^{\ast }$.

Define ${\mathrm{ad}}_X^{\ast }:={\mathrm{ad}}^{\ast }(X)$. We see that $${\mathrm{ad}}_X^{\ast }(\mu )=-<\mu ,[X,-]>,$$ for any $X$ in $\mathfrak{g}$ and $\mu$ in ${\mathfrak{g}}^{\ast }$.

3.3 Tangent Space of a Stabilizer of the Coadjoint Representation

For an element $\mu$ in $\mathfrak{g}$, the stabilizer $G_{\mu }$ of the coadjoint representation ${\mathrm{Ad}}^{\ast }$ of $G$ is $$G_{\mu }=\{g\in G\mid {\mathrm{Ad}}_g^{\ast }(\mu )=\mu \}.$$

We know that $G_{\mu }\subset G$ is a Lie subgroup. Let ${\mathfrak{g}}_{\mu }$ be the Lie algebra associated to $G_{\mu }$. Then $${\mathfrak{g}}_{\mu }=\{X\in \mathfrak{g}\mid {\mathrm{Ad}}_{\exp (tX)}^{\ast }\mu =\mu \}.$$ For a proof, see Proposition 9.1.13 in (Marsden and Ratiu 1999).

Differentiate the identity at $t=0$, we find that $${\mathfrak{g}}_{\mu }=\{X|{\mathrm{ad}}_X^{\ast }\mu =0\}.$$

4 Coadjoint Oribts

Let $G$ be a Lie group and $\mathfrak{g}$ be the associated Lie algebra. Given an element $\mu$ in the dual ${\mathfrak{g}}^{\ast }$ of the Lie algebra $\mathfrak{g}$, we denote by ${\mathcal{O}}_{\mu }$ the $G$-orbit of $\mu$ in ${\mathfrak{g}}^{\ast }$.

On a coadjoint orbit, there is a naturally defined symplectic structure. Consider the antisymmetric bilinear form ${\omega }_{\mu }$ defined by $${\omega }_{\mu }({\mathrm{ad}}_X^{\ast }\mu ,{\mathrm{ad}}_Y^{\ast }\mu )=-<\mu ,[X,Y]>,$$ where $X$ and $Y$ are vector field in $\mathfrak{g}$.

Theorem 1 Let $G$ be a Lie group and ${\mathcal{O}}_{\mu }\subset {\mathfrak{g}}^{\ast }$ be a coadjoint orbit. Then the antisymmetric bilinear form ${\omega }_{\mu }$ is a ${\mathrm{Ad}}_{g^{-1}}^{\ast }$-invariant symplectic form on ${\mathcal{O}}_{\mu }$.

For a proof of the theorem, see for example Theorem 14.3.1 in (Marsden and Ratiu 1999).

Corollary 1 The coadjoint orbit of a finite-dimensional Lie groups is of even dimension.

Via the diffeomorphism $\rho :G/G_{\mu }\to {\mathcal{O}}_{\mu }$, the manifold $G/G_{\mu }$ admits symplectic structure that pulls back ${\omega }_{\mu }$.

References

Audin, Michèle. 2012. Torus Actions on Symplectic Manifolds. Softcover reprint of the original 2nd ed. 2004 edition. Basel: Birkhäuser.

Gorbatsevich, V. V., A. L. Onishchik, and E. B. Vinberg. 1993. Lie Groups and Lie Algebras I: Foundations of Lie Theory Lie Transformation Groups. Edited by A. L. Onishchik. Encyclopaedia of Mathematical Sciences, Lie Groups and Lie Algebras. Berlin Heidelberg: Springer-Verlag. https://doi.org/10.1007/978-3-642-57999-8.

Lee, John M. 2012. Introduction to Smooth Manifolds. Vol. 218. Graduate Texts in Mathematics. New York, NY: Springer New York. https://doi.org/10.1007/978-1-4419-9982-5.

Marsden, Jerrold E., and Tudor S. Ratiu. 1999. Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems. 2nd edition. New York: Springer.