Lie Groups and Their Representations¶

Job: pass between Lie groups and Lie algebras via $\operatorname{Lie}$ and $\exp$; compact groups and the Peter–Weyl theorem as the analytic counterpart of complete reducibility.

Core¶

Lie groups and the Lie algebra functor¶

A Lie group $G$ is a smooth (or real/complex analytic) manifold with group operations smooth. The Lie algebra $$ \mathfrak{g}=T_e G \cong {\text{left-invariant vector fields on }G} $$ carries the Lie bracket of vector fields (or $[X,Y]=\frac{d}{dt}|_0\operatorname{Ad}_{\gamma(t)}Y$ for curves $\gamma$ with $\gamma'(0)=X$).

Functor $\operatorname{Lie}$. Smooth homomorphisms $\varphi:G\to H$ induce $\mathrm{d}\varphi_e:\mathfrak{g}\to\mathfrak{h}$ Lie algebra maps. For connected simply connected $G$, continuous/Lie homs $G\to H$ biject with Lie homs $\mathfrak{g}\to\mathfrak{h}$ (under standard hyp. on $H$).

Example	$G$	$\mathfrak{g}$
$\operatorname{GL}_n(\mathbb{R})$	invertible matrices	$\mathfrak{gl}_n=M_n$
$\operatorname{SL}_n$	det $1$	$\mathfrak{sl}_n=\operatorname{tr}0$
$\operatorname{SU}(n)$	unitary det $1$	$\mathfrak{su}(n)$ skew-Hermitian tr $0$
$\operatorname{SO}(n)$	orthogonal det $1$	$\mathfrak{so}(n)$ skew-symmetric
$S^1=U(1)$	circle	$i\mathbb{R}$

Adjoint representations. $\operatorname{Ad}:G\to\operatorname{GL}(\mathfrak{g})$, $\operatorname{Ad}_g=T_e(C_g)$, $C_g(x)=gxg^{-1}$; $\operatorname{ad}=\mathrm{d}(\operatorname{Ad})_e:\mathfrak{g}\to\mathfrak{gl}(\mathfrak{g})$, $\operatorname{ad}_x(y)=[x,y]$.

Exponential map¶

$\exp:\mathfrak{g}\to G$, for matrix groups $\exp(X)=\sum X^k/k!$; abstractly flow of left-invariant vector field at time $1$.

Property	Statement
Local diffeomorphism	$\mathrm{d}\exp_0=\mathrm{id}$; exp diffeomorphism near $0$ onto nbhd of $e$
One-parameter subgroups	$\gamma(t)=\exp(tX)$ all smooth homs $(\mathbb{R},+)\to G$
Not always surjective	$\exp:\mathfrak{sl}_2(\mathbb{R})\to\operatorname{SL}_2(\mathbb{R})$ misses matrices with trace $<-2$ in some components — actually fails surjectivity
Compact connected	$\exp$ surjective
$\operatorname{Ad}_{\exp X}=e^{\operatorname{ad}X}$	fundamental identity

Baker–Campbell–Hausdorff. Locally $\log(e^Xe^Y)=X+Y+\frac{1}{2}[X,Y]+\cdots$ — multiplies group near $e$ from Lie brackets alone.

From group reps to Lie algebra reps¶

Finite-dimensional smooth (equiv. continuous) representation $\rho:G\to\operatorname{GL}(V)$ differentiates to $\mathrm{d}\rho:\mathfrak{g}\to\mathfrak{gl}(V)$. For $G$ connected, $\rho$ determined by $\mathrm{d}\rho$; for $G$ connected simply connected, every f.d. $\mathfrak{g}$-rep integrates to a unique $G$-rep.

Complexification. Compact real form $\mathfrak{k}$ of complex semisimple $\mathfrak{g}_\mathbb{C}$: f.d. complex reps of $K$ $\leftrightarrow$ f.d. $\mathfrak{g}_\mathbb{C}$-modules $\leftrightarrow$ holomorphic reps of complex group (with care about centers / algebraic groups).

Compact groups: complete reducibility and Peter–Weyl¶

$G$ compact Hausdorff topological group (e.g. compact Lie): Haar measure $\mu$ bi-invariant (up to scaling).

Complete reducibility. Every continuous f.d. rep is unitary (average any Hermitian metric) and completely reducible (orthogonal complement of invariant subspace is invariant).

Matrix coefficients. For unitary irrep $(\rho,V)$, vectors $u,v\in V$: $g\mapsto\langle\rho(g)u,v\rangle$.

Theorem (Peter–Weyl). For $G$ compact: 1. Irreducible continuous unitary reps are finite-dimensional; matrix coefficients (suitably normalized) form an orthonormal system in $L^2(G)$. 2. Algebraic direct sum of all matrix coefficients is dense in $C(G)$ (uniform) and in $L^2(G)$. 3. Regular representation on $L^2(G)$ decomposes as $\bigoplus_{\pi\in\widehat{G}} \pi^{\oplus\dim\pi}$ (Hilbert sum).

Moral: $L^2(G)\cong\bigoplus_\pi \operatorname{End}(V_\pi)$ as Hilbert $G\times G$-reps — nonabelian Fourier series. Abelian case: classical Fourier on $S^1$ or finite abelian groups.

Weyl integration / character formula. Conjugacy classes meet maximal torus $T$; integration reduces to $T$ with Weyl denominator. Weyl character formula for compact connected semisimple $G$ matches the algebraic formula for $L(\lambda)$.

Homogeneous spaces (pointer)¶

Closed $H\le G$: $G/H$ smooth manifold. Induced reps $\operatorname{Ind}_H^G$ and Borel–Weil: holomorphic line bundles on $G/B$ ($B$ Borel) realize irreps as $H^0(G/B,\mathcal{L}_\lambda)$ — bridge to algebraic geometry.

See it¶

Spherical harmonics. $G=\operatorname{SO}(3)$, $H=\operatorname{SO}(2)$: $L^2(S^2)$ decomposes into irreps $V_{2\ell+1}$ of dimension $2\ell+1$ (spin $\ell$), realized as harmonic homogeneous polynomials of degree $\ell$. Laplace–Beltrami eigenvalues labeled by representation theory — prototype of spectral geometry on symmetric spaces.

Tannaka–Krein reconstruction. The tensor category of f.d. continuous reps of compact $G$ recovers $G$ as tensor-preserving automorphisms of the forgetful functor — conceptual inverse to “representation theory of $G$.” Algebraic-group analogue: Tannakian formalism for affine group schemes.

Check¶

Why does a Lie algebra hom $\mathfrak{g}\to\mathfrak{gl}(V)$ not always integrate to $G\to\operatorname{GL}(V)$ if $G$ is connected but not simply connected?

Kernel of $\widetilde{G}\to G$ (universal cover) may act nontrivially in the integrated rep of $\widetilde{G}$. Example: $\operatorname{Spin}(n)\to\operatorname{SO}(n)$; half-spin reps of $\mathfrak{so}(n)$ do not factor through $\operatorname{SO}(n)$. Projective reps of $G$ $\leftrightarrow$ linear reps of covers.

State why $\exp:M_n(\mathbb{C})\to\operatorname{GL}_n(\mathbb{C})$ is surjective but the real $\exp:M_n(\mathbb{R})\to\operatorname{GL}_n(\mathbb{R})$ is not.

Complex: Jordan form + $\exp$ on eigenvalues $\mathbb{C}\to\mathbb{C}^\times$ surjective. Real: matrices with negative eigenvalues and nontrivial Jordan structure for those eigenvalues need not be in the image (e.g. $\begin{pmatrix}-1&1\\0&-1\end{pmatrix}$ not an exponential in $M_2(\mathbb{R})$).

Pitfalls¶

Smooth vs continuous reps of Lie groups: automatic smoothness for f.d. continuous reps; infinite-dim unitary reps need domain care (unbounded Lie algebra operators).
Complex vs real Lie groups: $\operatorname{SL}_2(\mathbb{C})$ as real $6$-manifold vs complex $3$-fold — different $\operatorname{Lie}$ dimensions over $\mathbb{R}$.
Peter–Weyl needs compactness: noncompact semisimple groups have continuous series, not only discrete $L^2$-summands (Plancherel is more subtle).
Algebraic vs Lie group homs: polynomial maps $\operatorname{GL}_n\to\operatorname{GL}_m$ vs smooth; rational representations of algebraic groups $\neq$ all smooth reps of real points.

Refs¶

Hall, Lie Groups, Lie Algebras, and Representations
Brocker–tom Dieck, Representations of Compact Lie Groups
Knapp, Lie Groups Beyond an Introduction
Fulton–Harris, Representation Theory

Next¶

Foundations · Lie algebras · RT index · Algebraic topology · Algebraic geometry · Category theory · Higher Teichmüller theory — Anosov / Hitchin representations into semisimple Lie groups

Example	\(G\)	\(\mathfrak{g}\)
\(\operatorname{GL}_n(\mathbb{R})\)	invertible matrices	\(\mathfrak{gl}_n=M_n\)
\(\operatorname{SL}_n\)	det \(1\)	\(\mathfrak{sl}_n=\operatorname{tr}0\)
\(\operatorname{SU}(n)\)	unitary det \(1\)	\(\mathfrak{su}(n)\) skew-Hermitian tr \(0\)
\(\operatorname{SO}(n)\)	orthogonal det \(1\)	\(\mathfrak{so}(n)\) skew-symmetric
\(S^1=U(1)\)	circle	\(i\mathbb{R}\)