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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 \(\Psi: G\to \Diff(X)\) in the sense that \(\psi: G\times X\to X\), \(\psi(g, x)=\Psi_g(x)\) is a smooth map, where \(\Diff(X)\) is the group of diffeomorphisms of \(X\) and \(\Psi_g:=\Psi(g)\).

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: \[ \begin{aligned} \omicron_x: G&\to X\\ g&\mapsto g\cdot x:=\Psi_g(x). \end{aligned} \] This map \(\omicron_x\) 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 \(\omicron_x\): \[ \Orb_x=\omicron_x(G)=\{g\cdot x\vert g\in G \}, \] where \(g\cdot x:=\Psi_g(x)\).

The stabilizer of \(x\) in \(X\) is the preimage of \(\omicron\) \[ G_x=\omicron_x^{-1}(x)=\{g\vert g\cdot x=x\}\subset G. \]

Since the action \(\Psi\) is smooth, the stabilizer \(G_x=\omicron_x^{-1}(x)\) 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 \[ \begin{aligned} (\omicron_x\circ L_g)(h)=&\omicron_x(gh)=\Psi_{gh}(x)\\ =&(\Psi_g\circ\Psi_h)(x)=\Psi_g(\Psi_h(x))\\ =&\Psi_g(\omicron_x(h))=(\Psi_g\circ\omicron_x)(h). \end{aligned} \]

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

However, if \(G\) is compact, then \(\Orb_x\) 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 \(\Lg/\Lg_x=T_e(G)/T_e(G_x)\).

Induced by the orbit map \(\omicron_x\), we get a bijection \[ \begin{aligned} G/G_x&\to \Orb_x\\ gG_x&\mapsto g\cdot x. \end{aligned} \] The \(G\)-action on \(X\) induces a \(G\)-action on \(\Orb_x\). Moreover, the \(G\)-action on \(\Orb_x\) is transitive with the stabilizer \(G_x\). Hence \(G/G_x\) is diffeomorphic to \(\Orb_x\) (Theorem 3.3 in (Gorbatsevich, Onishchik, and Vinberg 1993)).

1.1 Conjugation Action

Consider the map \(\Psi: G\to\Aut(G)\) sending an element \(g\) to the inner automorphism \(\Psi_g\), i.e. \(\Psi_g(h)=ghg^{-1}\). Then \(\Psi\) is a group homomorphism. Indeed, for any \(x, g, h\in G\), we have \[ \Psi(gh)(x)=(gh)x(gh)^{-1}=g(hxh^{-1})g^{-1}=\Psi_g\Psi_h(x). \] Therefore, \(\Psi(gh)=\Psi_g\Psi_h\).

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

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 \[\Pi: G\to \operatorname{GL}(V)\] that is continuous in the sense that \(G\times V\to V\), \((g, v)\mapsto \Pi(g)(v)\) 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 \(T_eG\) 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 \(\Lg\), is called the Lie algebra associated to \(G\).

For each \(g\) in \(G\), we define \(\Ad_g=(\d\Psi _{g})_{e}: \Lg \to \Lg\) to be the derivative of the inner automorphism \(\Psi_g: G\to G\) at the origin. Let \(\exp(tX)\) be the unique integral curve associated to \(X\). Then \[ \begin{aligned} \Ad_g(X)=&(\d\Psi_g)_e(X)\\ =&\frac{\d}{\d\, t}\bigg\vert_{t=0}(\Psi_g(\exp(tX)))\\ =&\frac{\d}{\d\, t}\bigg\vert_{t=0}(g\exp(tX)g^{-1}). \end{aligned} \]

As the exponential map \(\exp: \Lg\to G\) commutes with Lie group homomorphism and its derivative, we see that \[ \exp(\Ad_g(X))=g\exp(X)g^{-1} \]

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 \[ \Ad_g([X, Y])=[\Ad_g(X), \Ad_g(Y)] \] which means that \(\Ad_g\) is a Lie algebra automorphism of \(\Lg\).

Consider the map \[ \begin{aligned} \Ad: G &\to \Aut(\Lg)\\ g &\mapsto \Ad_g. \end{aligned} \] For any \(g\) and \(h\) in \(G\), we have \[ \begin{aligned} \Ad(gh)=&\Ad_{gh}=(\d\Psi_{gh})_e\\ =&(\d(\Psi_g\circ\Psi_h))_e=(\d\Psi_g)_{\Psi_h(e)}\circ(\d\Psi_h)_e\\ =&(\d\Psi_g)_{e}\circ(\d\Psi_h)_e=\Ad(g)\Ad(h). \end{aligned} \] Hence, \(\Ad\) is a group representation called the adjoint representation of \(G\).

2.2 Coadjoint Representation of a Lie Group

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

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 \(\Lg\).

3.1 Adjoint Representation of a Lie Algebra

Consider the differential \((\d\Ad)_e\) of the adjoint representation of \(G\). For any \(X\) and \(Y\) in \(\Lg\), we have \[ \begin{aligned} &(\d\Ad)_e(X)(Y)\\ =&\frac{\d}{\d s}\bigg\vert_{s=0}(\Ad_{\exp(sX)}(Y))\\ =&\frac{\d}{\d s}\bigg\vert_{s=0}\left(\frac{\d}{\d t}\bigg\vert_{t=0}\exp(sX)\exp(tY)\exp(-sX)\right)\\ =&\frac{\d}{\d s}\bigg\vert_{s=0}\left(\exp(sX)Y\exp(-sX)\right)\\ =&(\exp(sX)XY\exp(-sX)-\exp(sX)Y\exp(-sX))X\big\vert_{s=0}\\ =&[X, Y]. \end{aligned} \]

We denote \((\d\Ad)_e\) as \(\ad\) and \(\ad(X)\) as \(\ad_X\). It can be checked that \(\ad([X, Y])=[\ad(X),\ad(Y)]\). Indeed, for any \(Z\), we have \[ \begin{aligned} \ad([X, Y])(Z)=&[[X, Y], Z]\\ =&[X, [Y, Z]]-[Y, [X, Z]]\\ =&\ad_X\ad_Y(Z)-\ad_Y\ad_X(Z). \end{aligned} \] Therefore, \(\ad:\Lg\to \Lgl(\Lg)\) defines a Lie algebra representation which is called the adjoint representation of the Lie algebra \(\Lg\).

3.2 Coadjoint Representation of a Lie Algebra

From the definition of \(\Ad^*\), one can check that the differential \(\ad^*=(\d\Ad^*)_e: \Lg \to \Lgl(\Lg^*)\) is a representation of \(\Lg\) in \(\Lg^*\), which is called the coadjoint representation of \(\Lg^*\).

Define \(\ad_X^*:=\ad^*(X)\). We see that \[ \ad_X^*(\mu)=-<\mu, [X, -]>, \] for any \(X\) in \(\Lg\) and \(\mu\) in \(\Lg^*\).

3.3 Tangent Space of a Stabilizer of the Coadjoint Representation

For an element \(\mu\) in \(\Lg\), the stabilizer \(G_\mu\) of the coadjoint representation \(\Ad^*\) of \(G\) is \[ G_\mu=\{g\in G\mid \Ad_g^*(\mu)=\mu\}. \]

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

Differentiate the identity at \(t=0\), we find that \[\Lg_\mu=\{X\vert \ad_X^*\mu=0\}.\]

4 Coadjoint Oribts

Let \(G\) be a Lie group and \(\Lg\) be the associated Lie algebra. Given an element \(\mu\) in the dual \(\Lg^*\) of the Lie algebra \(\Lg\), we denote by \(\O_\mu\) the \(G\)-orbit of \(\mu\) in \(\Lg^*\).

On a coadjoint orbit, there is a naturally defined symplectic structure. Consider the antisymmetric bilinear form \(\omega_\mu\) defined by \[ \omega_\mu(\ad_X^*\mu, \ad_Y^*\mu)=-<\mu, [X, Y]>, \] where \(X\) and \(Y\) are vector field in \(\Lg\).

Theorem 1 Let \(G\) be a Lie group and \(\O_\mu\subset\Lg^*\) be a coadjoint orbit. Then the antisymmetric bilinear form \(\omega_\mu\) is a \(\Ad_{g^{-1}}^*\)-invariant symplectic form on \(\O_\mu\).

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 \(\rho: G/G_\mu\to \O_\mu\), 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.