These notes synthesize material from multiple sources. Primary texts — Brian C. Hall, Lie Groups, Lie Algebras, and Representations (2nd ed., Springer GTM 222); Mark R. Sepanski, Compact Lie Groups (Springer GTM 235); James E. Humphreys, Introduction to Lie Algebras and Representation Theory (Springer GTM 9); Theodore Bröcker and Tammo tom Dieck, Representations of Compact Lie Groups (Springer GTM 98). Supplementary texts — N. Bourbaki, Éléments de Mathématique: Groupes et Algèbres de Lie; Joachim Hilgert and Karl-Hermann Neeb, Structure and Geometry of Lie Groups; Jacques Faraut, Analysis on Lie Groups. Online resources — Eckhard Meinrenken’s lecture notes (University of Toronto); Peter Woit’s notes on Lie Groups and Representations (Columbia University); lecture notes from MIT OCW 18.755. Background topology and manifold theory from James R. Munkres, Topology; Frank W. Warner, Foundations of Differentiable Manifolds and Lie Groups; William M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry.

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Chapter 1: Smooth Manifolds

Lie groups are simultaneously groups and smooth manifolds, with the group operations smooth maps. To state this precisely and work with it effectively, we need the language of differential geometry. This chapter develops the theory of smooth manifolds — charts, tangent spaces, vector fields, and the Lie bracket — at the level needed throughout the course. Readers with a solid background in differential geometry may skim this material, but the notation and conventions established here will be used without comment later.

1.1 Topological and Smooth Manifolds

A topological space \(M\) is an \(n\)-dimensional topological manifold if it is Hausdorff, second-countable, and locally homeomorphic to \(\mathbb{R}^n\): every point \(p \in M\) has an open neighbourhood \(U\) homeomorphic to an open subset of \(\mathbb{R}^n\). The Hausdorff condition prevents two distinct points from being topologically indistinguishable; second-countability guarantees the existence of partitions of unity, without which global constructions on manifolds become impossible.

Maximality is a bookkeeping convenience: any smooth atlas extends uniquely to a maximal one by including all charts compatible with those already present. So to specify a smooth structure, it suffices to exhibit any smooth atlas.

The fundamental examples: \(\mathbb{R}^n\) itself (one chart, the identity); the \(n\)-sphere \(S^n \subset \mathbb{R}^{n+1}\) (two stereographic projection charts, whose transition map is \(x \mapsto x/|x|^2\), smooth on \(\mathbb{R}^n \setminus \{0\}\)); the \(n\)-torus \(\mathbb{T}^n = \mathbb{R}^n / \mathbb{Z}^n\) (charts from local inverses of the quotient map); any open subset of \(\mathbb{R}^n\) (restriction of the identity chart); and, crucially for us, \(GL(n, \mathbb{R})\) as an open subset of \(M_n(\mathbb{R}) \cong \mathbb{R}^{n^2}\) (the determinant is a continuous function, so \(GL(n, \mathbb{R}) = \det^{-1}(\mathbb{R} \setminus \{0\})\) is open).

Smoothness is a local condition, well-defined because transition maps are smooth. Composition preserves smoothness, so smooth manifolds and smooth maps form a category.

1.2 Tangent Vectors and the Differential

The tangent space \(T_p M\) at a point \(p \in M\) formalises the idea of a “direction of motion” through \(p\). There are two useful equivalent definitions.

The geometric definition: a tangent vector at \(p\) is an equivalence class of smooth curves \(\gamma : (-\varepsilon, \varepsilon) \to M\) with \(\gamma(0) = p\), where \(\gamma_1 \sim \gamma_2\) if \((\varphi \circ \gamma_1)'(0) = (\varphi \circ \gamma_2)'(0)\) in some (equivalently any) chart \(\varphi\) near \(p\).

The algebraic definition is more intrinsic. A smooth function near \(p\) is a germ: a smooth \(f : U \to \mathbb{R}\) on some open \(U \ni p\). Let \(C^\infty_p\) denote the algebra of germs.

In a local chart \((U, \varphi)\) with coordinates \(x^1, \ldots, x^n\), the operators \(\partial/\partial x^i\big|_p\) defined by

\[ \frac{\partial}{\partial x^i}\bigg|_p f = \frac{\partial(f \circ \varphi^{-1})}{\partial x^i}\bigg|_{\varphi(p)} \]

form a basis for \(T_p M\), which therefore has dimension \(n = \dim M\). Every \(v \in T_p M\) writes uniquely as \(v = \sum_i v^i \partial/\partial x^i|_p\). The two definitions are equivalent: the curve \(\gamma\) with \((\varphi \circ \gamma)'(0) = (v^1, \ldots, v^n)\) represents \(v\) as a derivation via \(v(f) = (f \circ \gamma)'(0)\).

The rank of \(f\) at \(p\) is \(\operatorname{rank}(df_p)\). A map \(f : M \to N\) is an immersion if \(df_p\) is injective for all \(p\) (so \(\dim M \leq \dim N\)), and a submersion if \(df_p\) is surjective for all \(p\).

This theorem will be used repeatedly: to show that \(SL(n, \mathbb{R})\), \(O(n)\), and the other classical groups are smooth submanifolds of \(GL(n, \mathbb{R})\).

1.3 Vector Fields and the Lie Bracket

A smooth vector field on \(M\) is a smooth assignment \(X : M \to TM\) with \(X(p) \in T_p M\). Equivalently, \(X\) is a derivation on \(C^\infty(M)\): an \(\mathbb{R}\)-linear map \(X : C^\infty(M) \to C^\infty(M)\) satisfying \(X(fg) = fX(g) + gX(f)\). In local coordinates, \(X = \sum_i a^i(x) \partial/\partial x^i\) with \(a^i \in C^\infty(U)\).

One must check this is well-defined (that \([X,Y]\) is again a derivation, not a second-order operator) and smooth. In local coordinates with \(X = \sum_i a^i \partial_i\) and \(Y = \sum_j b^j \partial_j\):

\[ X(Y(f)) = \sum_{i,j} a^i \frac{\partial b^j}{\partial x^i} \frac{\partial f}{\partial x^j} + \sum_{i,j} a^i b^j \frac{\partial^2 f}{\partial x^i \partial x^j}. \]

The second-order terms are symmetric in \(i, j\) and cancel in \(X(Y(f)) - Y(X(f))\), leaving

\[ [X, Y](f) = \sum_j \left(\sum_i a^i \frac{\partial b^j}{\partial x^i} - b^i \frac{\partial a^j}{\partial x^i}\right) \frac{\partial f}{\partial x^j}. \]

So \([X, Y] = \sum_j w^j \partial_j\) where \(w^j = \sum_i (a^i \partial_i b^j - b^i \partial_i a^j)\), confirming it is a first-order operator and hence a vector field.

These three properties make the space of smooth vector fields \(\mathfrak{X}(M)\) into a Lie algebra over \(\mathbb{R}\). The Jacobi identity can be verified directly from the definition (expand all three double brackets and observe cancellation), or more elegantly by noting that \(X \mapsto [X, -]\) is a derivation of the bracket: \([X, [Y, Z]] = [[X,Y], Z] + [Y, [X, Z]]\).

Naturality: if \(f : M \to N\) is a diffeomorphism, then \(f_*[X, Y] = [f_* X, f_* Y]\) (proved by unwinding the definition: \((f_*[X,Y])(g) = [X,Y](g \circ f) = X(Y(g \circ f)) - Y(X(g \circ f)) = [f_*X, f_*Y](g)\)). This naturality property is essential for proving that the Lie bracket of left-invariant vector fields on a Lie group is again left-invariant.

1.4 Submanifolds

The distinction matters: the figure-eight \(f : (-\pi, \pi) \to \mathbb{R}^2\) given by \(f(t) = (\sin t, \sin 2t)\) is an immersion whose image is not a regular submanifold (the image has a self-intersection). The line of irrational slope \(f : \mathbb{R} \to \mathbb{T}^2\), \(f(t) = (e^{it}, e^{i\alpha t})\) for irrational \(\alpha\), is an injective immersion whose image is dense in \(\mathbb{T}^2\), hence not an embedding.

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Chapter 2: Lie Groups and Lie Algebras

2.1 Lie Groups: Definition and Classical Examples

The requirement that \(G\) carry both structures simultaneously is far from vacuous: it forces an intricate interplay between the algebraic and geometric structures that is the central subject of this course.

The basic examples:

\(\mathbb{R}^n\) and \(\mathbb{C}^n\) under addition are Lie groups of dimensions \(n\) and \(2n\). The torus \(\mathbb{T}^n = (S^1)^n\) under componentwise multiplication is a compact Lie group of dimension \(n\). These are the simplest examples; the rich theory begins with the matrix groups.

The general linear group \(GL(n, \mathbb{R}) = \{A \in M_n(\mathbb{R}) : \det A \neq 0\}\) is a Lie group of dimension \(n^2\). It is an open submanifold of \(M_n(\mathbb{R}) \cong \mathbb{R}^{n^2}\) (since \(\det : M_n(\mathbb{R}) \to \mathbb{R}\) is continuous). Multiplication is a polynomial map in the matrix entries, hence smooth; inversion is \(A \mapsto (\det A)^{-1} \operatorname{Adj}(A)\), a rational function with nonzero denominator on \(GL(n, \mathbb{R})\), hence smooth.

Similarly \(GL(n, \mathbb{C})\) is a Lie group of real dimension \(2n^2\).

The special linear group \(SL(n, \mathbb{R}) = \ker(\det : GL(n, \mathbb{R}) \to \mathbb{R}^*)\) is the level set of the determinant map. The differential of \(\det\) at \(I\) in direction \(A\) is \(\operatorname{tr}(A)\) (compute \(d/dt|_{t=0} \det(I + tA) = \operatorname{tr}(A)\)), so \(\det : GL \to \mathbb{R}^*\) is a submersion and \(SL(n, \mathbb{R})\) is a regular submanifold of dimension \(n^2 - 1\).

The orthogonal group \(O(n) = \{A \in GL(n, \mathbb{R}) : A^T A = I\}\) is the level set of \(\varphi : GL(n, \mathbb{R}) \to \operatorname{Sym}(n)\), \(\varphi(X) = X^T X\). This map has constant rank: for any \(A \in GL\),

\[ \varphi(XA) = A^T X^T X A = L_{A^T}(R_A(\varphi(X))), \]

so \(D\varphi(XA) \cdot DR_A(X) = DL_{A^T}(\varphi(X)) \cdot DR_A(\varphi(X)) \cdot D\varphi(X)\). Since \(R_A\) and \(L_{A^T}\) are diffeomorphisms, \(\operatorname{rank}(D\varphi(XA)) = \operatorname{rank}(D\varphi(X))\) for all \(X\). By the rank theorem, \(O(n) = \varphi^{-1}(I)\) is a regular submanifold of dimension \(n^2 - n(n+1)/2 = n(n-1)/2\).

The special orthogonal group \(SO(n) = O(n) \cap SL(n, \mathbb{R})\) consists of the orientation-preserving orthogonal matrices; it is an open submanifold of \(O(n)\) (since \(\det : O(n) \to \{\pm 1\}\) is locally constant).

The unitary group \(U(n) = \{A \in GL(n, \mathbb{C}) : A^* A = I\}\) and special unitary group \(SU(n) = U(n) \cap SL(n, \mathbb{C})\) are defined analogously; they are compact Lie groups of real dimensions \(n^2\) and \(n^2 - 1\).

The symplectic group (compact form) \(Sp(n) = \{A \in GL(n, \mathbb{H}) : A^* A = I\}\), where \(\mathbb{H}\) is the quaternions and \(A^*\) is the conjugate transpose, is a compact Lie group of real dimension \(n(2n+1)\).

2.2 Lie Group Homomorphisms and Constant Rank

This immediately implies the rank theorem applies globally: kernels and images of Lie group homomorphisms are regular submanifolds (hence Lie subgroups).

2.3 Left-Invariant Vector Fields and the Lie Bracket

The key to extracting an algebraic structure from a Lie group is to use the group structure to propagate tangent vectors globally.

Left-invariant vector fields are completely determined by their value at the identity: if \(X_e = A \in T_e G\), then \(X_a = d(\ell_a)_e(A)\) for all \(a \in G\). Conversely, given any \(A \in T_e G\), the prescription \(X_a = d(\ell_a)_e(A)\) defines a smooth left-invariant vector field (smoothness follows because \((a, v) \mapsto d(\ell_a)_e(v)\) is smooth in both arguments). This gives a vector space isomorphism

\[ \{\text{left-invariant vector fields on } G\} \xrightarrow{\;\;\sim\;\;} T_e G, \quad X \mapsto X_e. \]

The Lie bracket of two left-invariant vector fields is left-invariant (by naturality of the Lie bracket under \(\ell_a\)). So the Lie bracket restricts to a bracket on the finite-dimensional vector space \(T_e G\).

For matrix Lie groups, this bracket is the matrix commutator. We now derive this explicitly.

2.4 Lie Algebras of the Classical Groups

The Lie algebra of a matrix Lie group \(G \subseteq GL(n, \mathbb{F})\) is characterised by

\[ \mathfrak{g} = T_I G = \{A \in M_n(\mathbb{F}) : e^{tA} \in G \text{ for all } t \in \mathbb{R}\}. \]

(The second equality will be proved once we have the exponential map; for now we compute directly from the first.)

Differentiating the defining equations of each classical group at the identity:

• \(\mathfrak{gl}(n, \mathbb{R}) = M_n(\mathbb{R})\) (no constraint).
• \(\mathfrak{sl}(n, \mathbb{R}) = \{A \in M_n(\mathbb{R}) : \operatorname{tr}(A) = 0\}\). Since \(\det(e^{tA}) = e^{t \operatorname{tr}(A)}\), this equals 1 for all \(t\) iff \(\operatorname{tr}(A) = 0\).
• \(\mathfrak{o}(n) = \mathfrak{so}(n) = \{A \in M_n(\mathbb{R}) : A^T + A = 0\}\). Differentiate \((e^{tA})^T e^{tA} = I\) to get \(A^T + A = 0\). Conversely, if \(A^T = -A\) then \((e^{tA})^T = e^{tA^T} = e^{-tA} = (e^{tA})^{-1}\), so \(e^{tA} \in O(n)\). Dimension: \(n(n-1)/2\).
• \(\mathfrak{u}(n) = \{A \in M_n(\mathbb{C}) : A^* + A = 0\}\). Differentiate \((e^{tA})^* e^{tA} = I\). Dimension: \(n^2\) (over \(\mathbb{R}\)).
• \(\mathfrak{su}(n) = \{A \in M_n(\mathbb{C}) : A^* + A = 0,\ \operatorname{tr}(A) = 0\}\). Dimension: \(n^2 - 1\).
• \(\mathfrak{sp}(n)\) (compact form) consists of quaternion-skew matrices; has real dimension \(n(2n+1)\).

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Chapter 3: The Exponential Map

3.1 One-Parameter Subgroups

A one-parameter subgroup of a Lie group \(G\) is a smooth group homomorphism \(\varphi : (\mathbb{R}, +) \to G\). The defining equations are smoothness and \(\varphi(s + t) = \varphi(s)\varphi(t)\) for all \(s, t \in \mathbb{R}\), with \(\varphi(0) = e\). Setting \(A = \varphi'(0) \in \mathfrak{g}\), we claim that the tangent vector at the identity determines the entire homomorphism.

3.2 The Exponential Map

By rescaling: \(\varphi_A(t) = \varphi_{tA}(1) = \exp(tA)\). So the one-parameter subgroup generated by \(A\) is \(t \mapsto \exp(tA)\), and \(\frac{d}{dt}\exp(tA)\big|_{t=0} = A\).

For matrix Lie groups \(G \subseteq GL(n, \mathbb{F})\), the matrix exponential

\[ e^A = \sum_{k=0}^\infty \frac{A^k}{k!} = I + A + \frac{A^2}{2!} + \frac{A^3}{3!} + \cdots \]

converges absolutely for every \(A \in M_n(\mathbb{F})\): if \(m = \max_{ij} |A_{ij}|\), then \(\max_{ij}|(A^\ell)_{ij}| \leq (nm)^\ell\) by induction, so \(\sum \|A^k/k!\| \leq \sum (nm)^k/k! = e^{nm} < \infty\). The map \(t \mapsto e^{tA}\) satisfies \(e^{(s+t)A} = e^{sA}e^{tA}\) (from \(e^Ae^B = e^{A+B}\) when \(AB = BA\)) and \((d/dt)e^{tA}\big|_{t=0} = A\), so \(e^{tA}\) is the one-parameter subgroup generated by \(A\), and \(\exp(A) = e^A\).

This shows \(e^A \in SL(n)\) iff \(\operatorname{tr}(A) = 0\), confirming \(\mathfrak{sl}(n) = \ker(\operatorname{tr})\).

3.3 The Baker-Campbell-Hausdorff Formula

When \(A\) and \(B\) commute, \(e^A e^B = e^{A+B}\). In general, the product \(e^A e^B = e^{C(A,B)}\) for some \(C(A,B)\) that depends on the Lie bracket. The Baker-Campbell-Hausdorff (BCH) formula expresses \(C\) as a formal power series in iterated brackets:

The key point is that \(C(A, B)\) lies in the Lie algebra generated by \(A\) and \(B\) under the bracket, not in the larger associative algebra. A clean proof uses the Dynkin formula \(C(A,B) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n} \sum_{\substack{p_i + q_i > 0 \\ 1 \leq i \leq n}} \frac{(\operatorname{ad} A)^{p_1}(\operatorname{ad} B)^{q_1} \cdots (\operatorname{ad} A)^{p_n}}{p_1! q_1! \cdots p_n! q_n! (p_n + q_n)} B\) (where terms with \(q_n = 0\) use \(A\) in place of \(B\)).

The BCH formula has a profound consequence: the local group structure of \(G\) near \(e\) is completely determined by its Lie algebra \(\mathfrak{g}\). Two Lie groups have the same local structure (are locally isomorphic) if and only if their Lie algebras are isomorphic. This is why the algebraic theory of Lie algebras captures so much about the geometry of Lie groups.

3.4 Naturality and the Exponential Diagram

This theorem is indispensable: it means homomorphisms of Lie groups correspond exactly to linear maps between their Lie algebras that preserve the bracket, at least for simply connected groups (the precise statement is Theorem 4.4).

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Chapter 4: Connectedness and Covering Theory

4.1 Connected Components

The Frobenius correspondence. A Lie subalgebra \(\mathfrak{h} \subseteq \mathfrak{g}\) is a vector subspace closed under the bracket. Frobenius’ theorem (the involutive distribution theorem) implies that \(\mathfrak{h}\) determines a unique connected Lie subgroup: form the left-invariant distribution \(\mathcal{D}_a = d(\ell_a)_e(\mathfrak{h}) \subseteq T_a G\); since \(\mathfrak{h}\) is closed under brackets, \(\mathcal{D}\) is involutive; Frobenius gives a unique maximal connected integral submanifold \(H\) through \(e\), and one verifies \(H\) is a subgroup. The result is:

4.2 Covering Spaces and the Universal Cover

A smooth map \(p : \tilde{M} \to M\) is a covering map if every \(x \in M\) has an open neighbourhood \(U\) such that \(p^{-1}(U)\) is a disjoint union of open sets each mapped diffeomorphically onto \(U\). The universal cover \(\tilde{M}\) of \(M\) is the unique simply connected covering space (unique up to isomorphism).

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Chapter 5: Fundamental Groups of the Classical Lie Groups

5.1 Deformation Retracts

Computing \(\pi_1\) of the classical groups requires identifying them up to homotopy equivalence with more familiar spaces. The key tool is the Gram-Schmidt orthogonalisation process.

The deformation retract for real matrices is constructed in three steps. Given \(A \in SL(n, \mathbb{R})\) with columns \(u_1, \ldots, u_n\):

1. Upper-triangularise: for each pair \(i < j\), subtract \(\frac{\langle u_j, u_i \rangle}{\|u_i\|^2} u_i\) from \(u_j\), making columns orthogonal; this is a path in \(SL(n, \mathbb{R})\) (the transformation is upper-triangular with 1s on the diagonal).
2. Normalise: scale each column \(u_k \mapsto u_k/\|u_k\|\); this is a path in \(SL(n, \mathbb{R})\) (adjust the last column to maintain determinant 1).
3. Reach \(SO(n)\): the result has orthonormal columns, so lies in \(O(n)\cap SL(n,\mathbb{R}) = SO(n)\).

Each step is continuous in \(A\), giving a continuous retraction \(r : SL(n, \mathbb{R}) \to SO(n)\) with \(r|_{SO(n)} = \mathrm{id}\).

5.2 Fundamental Groups of \(SO(n)\)

\(SO(1) = \{1\}\), so \(\pi_1 = 0\). \(SO(2) \cong S^1\) via \(R_\theta \leftrightarrow e^{i\theta}\), so \(\pi_1(SO(2)) = \mathbb{Z}\). For \(n = 3\): \(SO(3)\) is homeomorphic to \(\mathbb{R}P^3\) (identify a rotation with the pair (axis, angle \(\theta \in [0,\pi]\)), modulo the relation that a rotation by \(\pi\) around \(u\) equals a rotation by \(\pi\) around \(-u\)). Since \(\pi_1(\mathbb{R}P^3) = \mathbb{Z}/2\mathbb{Z}\), we get \(\pi_1(SO(3)) = \mathbb{Z}/2\mathbb{Z}\).

For general \(n \geq 3\), we use the fibre bundle

\[ SO(n) \hookrightarrow SO(n+1) \twoheadrightarrow S^n, \]

where \(SO(n+1)\) acts transitively on \(S^n\) (via \(A \cdot e_{n+1} = \) last column of \(A\)) with stabiliser of \(e_{n+1}\) being \(\{A : Ae_{n+1}=e_{n+1}\} \cong SO(n)\). The long exact homotopy sequence gives:

\[ \cdots \to \pi_2(S^n) \to \pi_1(SO(n)) \to \pi_1(SO(n+1)) \to \pi_1(S^n) \to \cdots \]

For \(n \geq 3\): \(\pi_2(S^n) = 0\) and \(\pi_1(S^n) = 0\), so \(\pi_1(SO(n)) \cong \pi_1(SO(n+1))\). By induction from \(\pi_1(SO(3)) = \mathbb{Z}/2\mathbb{Z}\): \(\pi_1(SO(n)) = \mathbb{Z}/2\mathbb{Z}\) for all \(n \geq 3\).

5.3 Fundamental Groups of \(SU(n)\) and \(Sp(n)\)

\(SU(2) \cong S^3\): Write \(SU(2) = \left\{\begin{pmatrix}a & -\bar{b}\\ b & \bar{a}\end{pmatrix} : |a|^2 + |b|^2 = 1\right\}\). The map \(\begin{pmatrix}a & -\bar{b}\\ b & \bar{a}\end{pmatrix} \mapsto (a, b) \in \mathbb{C}^2\) identifies \(SU(2)\) with \(S^3 = \{(a,b) \in \mathbb{C}^2 : |a|^2+|b|^2=1\}\). So \(\pi_1(SU(2)) = \pi_1(S^3) = 0\).

For \(n \geq 2\): the fibre bundle \(SU(n) \hookrightarrow SU(n+1) \twoheadrightarrow S^{2n+1}\) gives a long exact sequence with \(\pi_2(S^{2n+1}) = \pi_1(S^{2n+1}) = 0\) (for \(n \geq 1\)), so \(\pi_1(SU(n)) \cong \pi_1(SU(n+1))\). By induction from \(\pi_1(SU(1)) = 0\): \(\pi_1(SU(n)) = 0\) for all \(n \geq 1\). Similarly \(\pi_1(Sp(n)) = 0\) via the bundle \(Sp(n) \hookrightarrow Sp(n+1) \twoheadrightarrow S^{4n+3}\).

5.4 Spin Groups and the Map \(SU(2) \to SO(3)\)

Since \(\pi_1(SO(n)) = \mathbb{Z}/2\mathbb{Z}\) for \(n \geq 3\), the universal cover \(\mathrm{Spin}(n) \to SO(n)\) is a 2-sheeted covering. For \(n = 3\): since \(SU(2)\) is simply connected and there is a 2-to-1 Lie group homomorphism \(\Phi : SU(2) \to SO(3)\), we have \(\mathrm{Spin}(3) \cong SU(2)\).

The map \(\Phi\) is constructed as follows. Identify \(\mathbb{R}^3\) with the traceless skew-Hermitian matrices \(\mathfrak{su}(2) = \left\{\begin{pmatrix}it & -\bar{z}\\ z & -it\end{pmatrix} : t \in \mathbb{R},\, z \in \mathbb{C}\right\} \cong \mathbb{R}^3\) (with the Euclidean inner product \(\langle X, Y\rangle = -\frac{1}{2}\mathrm{tr}(XY)\)). For \(A \in SU(2)\), define \(\Phi(A)(X) = AXA^{-1} = AXA^*\). This is an isometry of \(\mathfrak{su}(2)\) with determinant 1 (since \(A \mapsto AXA^*\) is connected to the identity through \(e^{tB}\)), so \(\Phi(A) \in SO(3)\). The kernel is \(\{A : AXA^* = X \ \forall X\} = Z(SU(2)) = \{\pm I\}\). Since \(\dim SU(2) = \dim SO(3) = 3\) and \(\Phi\) has trivial differential kernel (as one checks), \(\Phi\) is a 2-to-1 surjective covering homomorphism.

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Chapter 6: Abelian Lie Groups and Maximal Tori

6.1 Abelian Lie Groups

6.2 The Closed Subgroup Theorem

One of the most useful structural theorems in Lie theory converts an algebraic condition into smoothness.

6.3 Maximal Tori and Cartan Subalgebras

The maximal tori in the classical compact Lie groups are:

• \(SO(2n)\): diagonal blocks \(T = \{\mathrm{diag}(R_{\theta_1},\ldots,R_{\theta_n})\}\), rank \(n\), where \(R_\theta = \bigl(\begin{smallmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{smallmatrix}\bigr)\).
• \(SO(2n+1)\): \(T = \{\mathrm{diag}(R_{\theta_1},\ldots,R_{\theta_n},1)\}\), rank \(n\).
• \(U(n)\): \(T = \{\mathrm{diag}(e^{i\theta_1},\ldots,e^{i\theta_n})\}\), rank \(n\).
• \(SU(n)\): same with \(\sum\theta_k = 0\), rank \(n-1\).
• \(Sp(n)\): \(T = \{\mathrm{diag}(e^{i\theta_1},\ldots,e^{i\theta_n},e^{-i\theta_1},\ldots,e^{-i\theta_n})\}\) (under \(Sp(n) \subset U(2n)\)), rank \(n\).

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Chapter 7: Structure Theory of Lie Algebras

The structure theory of Lie algebras — the classification of solvable, nilpotent, and semisimple algebras — is one of the great achievements of nineteenth and early twentieth century mathematics, largely due to Wilhelm Killing and Élie Cartan. For our purposes in representation theory, the most important results are Weyl’s theorem on complete reducibility and the classification of semisimple algebras by root systems. This chapter develops the algebraic foundations.

7.1 Solvable and Nilpotent Lie Algebras

Every nilpotent Lie algebra is solvable (since \(\mathfrak{g}^{(k)} \subseteq \mathfrak{g}^k\)). The paradigm examples: the algebra \(\mathfrak{b}_n\) of upper triangular \(n \times n\) matrices is solvable (its derived algebra is strictly upper triangular, which is nilpotent). The algebra \(\mathfrak{n}_n\) of strictly upper triangular matrices is nilpotent. The algebra \(\mathfrak{gl}(n)\) for \(n \geq 2\) is neither: \([\mathfrak{gl}(n), \mathfrak{gl}(n)] = \mathfrak{sl}(n)\) and \([\mathfrak{sl}(n), \mathfrak{sl}(n)] = \mathfrak{sl}(n)\).

Solvability and nilpotency are preserved under quotients and subalgebras. Extensions of solvable by solvable are solvable: if \(\mathfrak{h}\) is an ideal with \(\mathfrak{h}\) and \(\mathfrak{g}/\mathfrak{h}\) both solvable, then \(\mathfrak{g}\) is solvable.

7.2 Engel’s Theorem

7.3 Lie’s Theorem

7.4 The Killing Form and Cartan’s Criterion

The Killing form is \(\mathrm{ad}\)-invariant: \(B([X,Z], Y) + B(Z, [X,Y]) = 0\), i.e., \(B([X,Z],Y) = -B(Z,[X,Y])\). This follows from the trace identity \(\mathrm{tr}([A,B]C) = \mathrm{tr}(A[B,C])\) applied to \(\mathrm{ad}(X), \mathrm{ad}(Z), \mathrm{ad}(Y)\).

A Lie algebra is semisimple if its radical (the largest solvable ideal) is zero, equivalently if it has no nonzero abelian ideals.

7.5 Weyl’s Theorem and the Casimir Element

The commutation follows from the \(\mathrm{ad}\)-invariance of \(B_\rho\): \([\rho(X), C_\rho] = \sum_i [\rho(X), \rho(X_i)]\rho(X^i) + \rho(X_i)[\rho(X),\rho(X^i)] = \sum_i \rho([X,X_i])\rho(X^i) + \rho(X_i)\rho([X,X^i])\), which telescopes to zero when the basis transformation induced by \(\mathrm{ad}(X)\) is taken into account.

Chapter 8: Haar Measure and Integration on Lie Groups

The representation theory of compact Lie groups, which occupies the remainder of these notes, rests on a foundation of analysis: the existence of a canonical, translation-invariant measure on every compact Lie group. This chapter constructs that measure — the Haar measure — using the language of differential forms, establishes its key invariance properties, and records the integration identities that will drive all subsequent averaging arguments.

8.1 Differential Forms Review

We briefly recall the theory of differential forms, both to fix notation and to set up the construction of Haar measure.

A \(k\)-form on a smooth manifold \(M\) is a smooth section of the exterior power bundle \(\bigwedge^k T^*M\). In local coordinates \((x^1, \ldots, x^n)\), every \(k\)-form writes as

\[ \alpha = \sum_{I} a_I(x)\, dx^{i_1} \wedge \cdots \wedge dx^{i_k}, \]

where the sum runs over increasing multi-indices \(I = (i_1 < \cdots < i_k)\). We denote the space of smooth \(k\)-forms on \(M\) by \(\Omega^k(M)\).

The nilpotency \(d^2=0\) is the algebraic heart of de Rham cohomology.

An \(n\)-form on an oriented \(n\)-manifold can be integrated. In a positively oriented chart \((U, \varphi)\) with \(\varphi = (x^1,\ldots,x^n)\), if \(\operatorname{supp}(\omega) \subset U\), then

\[ \int_M \omega = \int_{\varphi(U)} a(x)\, dx^1 \cdots dx^n, \]

where \(\omega = a(x)\, dx^1 \wedge \cdots \wedge dx^n\). A volume form on an \(n\)-manifold is a nowhere-vanishing \(n\)-form; its existence is equivalent to orientability.

8.1.1 Left-Invariant and Right-Invariant Forms

Let \(G\) be a Lie group of dimension \(n\). For each \(a \in G\), we have the left-translation diffeomorphism \(\ell_a: G \to G\), \(\ell_a(x) = ax\), and the right-translation \(r_a: G \to G\), \(r_a(x) = xa\).

The space of left-invariant \(k\)-forms on \(G\) is in natural bijection with \(\bigwedge^k T_e^* G\): a left-invariant form is uniquely determined by its value at the identity, because

\[ \omega_a = (\ell_{a^{-1}})^* \omega_e \quad \text{for all } a \in G. \]

In particular, the space of left-invariant \(n\)-forms is one-dimensional (since \(\dim \bigwedge^n T_e^* G = 1\)).

8.2 Existence and Uniqueness of Haar Measure

8.3 Classical Group Examples

For \(G = \mathrm{U}(n)\), the maximal torus is \(T = \{\operatorname{diag}(e^{i\theta_1}, \ldots, e^{i\theta_n}) : \theta_k \in \mathbb{R}\}\), a copy of \((S^1)^n\). The Haar measure on \(T\) is

\[ d\mu_T = \frac{1}{(2\pi)^n}\, d\theta_1 \wedge \cdots \wedge d\theta_n. \]

Writing down the Haar measure on all of \(\mathrm{U}(n)\) explicitly requires the Weyl integration formula (Theorem 14.1), which expresses it in terms of the torus measure twisted by the Weyl denominator. For \(\mathrm{SU}(2)\), the Haar measure can be written in Euler angle coordinates, but this is most cleanly expressed via the integration formula of Chapter 14.

The most important consequence of Haar measure for representation theory is the averaging trick: given any positive-definite Hermitian form on a representation space, we average it over the group to obtain a \(G\)-invariant one. This is the engine behind Weyl’s unitary trick (Theorem 9.2).

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Chapter 9: Representation Theory — Foundations

With Haar measure in hand, we can develop the foundations of representation theory for compact Lie groups. The central results of this chapter — Schur’s Lemma and Weyl’s complete reducibility theorem — are the algebraic pillars supporting everything that follows.

9.1 Basic Definitions

A \(G\)-module isomorphism is a bijective intertwining operator; we write \(V \cong W\) when such an isomorphism exists. A \(G\)-module \(V\) is:

• irreducible (or a simple \(G\)-module) if it has no proper nonzero \(G\)-invariant subspaces;
• completely reducible if \(V = V_1 \oplus \cdots \oplus V_k\) as a direct sum of irreducible submodules.

The standard constructions of linear algebra carry \(G\)-module structures:

Note that \(\mathrm{Hom}(V, W) \cong V^* \otimes W\) as \(G\)-modules. In particular, \(\mathrm{End}(V) = V^* \otimes V\) and the algebra of \(G\)-endomorphisms is \(\mathrm{End}_G(V) = \mathrm{Hom}_G(V,V)\).

9.1.1 Lie Algebra Representations

Every Lie group representation \(\rho: G \to \mathrm{GL}(V)\) differentiates to a Lie algebra representation \(\rho_*: \mathfrak{g} \to \mathrm{End}(V)\), where \(\rho_*(X) = \frac{d}{dt}\big|_{t=0} \rho(e^{tX})\). When \(G\) is connected, \(\rho\) is determined by \(\rho_*\); when \(G\) is simply connected, every Lie algebra representation integrates to a unique Lie group representation.

9.2 Schur’s Lemma

Schur’s Lemma is the single most important structural result in representation theory, underlying every orthogonality and classification theorem that follows.

9.3 Complete Reducibility for Compact Groups

The compactness of \(G\) — via the Haar averaging trick — is precisely what ensures every representation is completely reducible.

9.4 Isotypical Decomposition

Given complete reducibility, we can decompose any representation into isotypical components — the “homogeneous pieces” corresponding to each isomorphism class of irreducibles.

Let \(\hat{G}\) denote a set of representatives for the isomorphism classes of irreducible finite-dimensional \(G\)-modules. For each \(\sigma \in \hat{G}\), let \(E_\sigma\) denote the corresponding irreducible module.

The orthogonality of distinct isotypical components — \(V_\sigma \perp V_\tau\) for \(\sigma \neq \tau\) — follows directly from Schur’s Lemma: any \(G\)-invariant map between non-isomorphic irreducibles is zero, so the projection onto \(V_\sigma\) kills all of \(V_\tau\).

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Chapter 10: The Peter-Weyl Theorem

The Peter-Weyl theorem is the central analytic theorem of the representation theory of compact Lie groups. It describes how the regular representation on \(L^2(G)\) decomposes, and shows that the matrix coefficients of irreducible representations form an orthonormal basis for \(L^2(G)\). This is the non-commutative generalisation of the classical Fourier theory on the circle.

10.1 Matrix Coefficients

Matrix coefficients are continuous functions on \(G\). As \(u, v\) vary over \(V\) and \(\rho\) varies over all irreducible representations, these functions span a dense subspace of \(C(G)\) (continuous functions on \(G\)). The Peter-Weyl theorem makes this precise in \(L^2\).

10.2 Schur Orthogonality Relations

The orthogonality of matrix coefficients from different irreducible representations is the key computational tool.

10.3 Characters

Characters enjoy several fundamental properties:

• Class function: \(\chi_V(bab^{-1}) = \chi_V(a)\) for all \(a, b \in G\) (trace is similarity-invariant).
• Additivity: \(\chi_{V \oplus W} = \chi_V + \chi_W\).
• Multiplicativity: \(\chi_{V \otimes W} = \chi_V \cdot \chi_W\).
• Duality: \(\chi_{V^*} = \overline{\chi_V}\).
• Isomorphism invariant: \(V \cong W \Rightarrow \chi_V = \chi_W\). (The converse also holds, as a corollary of the Peter-Weyl theorem.)

10.4 The Peter-Weyl Theorem

The regular representation of \(G\) on \(L^2(G)\) is the representation by left (or right) translations: \((\lambda(a)f)(x) = f(a^{-1}x)\). The Peter-Weyl theorem gives the complete decomposition of this representation.

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Chapter 11: Representations of \(\mathrm{SU}(2)\) and \(\mathfrak{sl}(2, \mathbb{C})\)

The representation theory of \(\mathrm{SU}(2)\) is both an important example in its own right and the prototype for the general theory of compact semisimple groups. By working through the \(\mathrm{SU}(2)\) case explicitly — using concrete polynomial models — we build the intuition needed for the general weight theory of Chapter 12 and the Cartan-Weyl classification of Chapter 13.

11.1 The Lie Algebra \(\mathfrak{sl}(2, \mathbb{C})\)

Recall from Chapter 5 that \(\mathfrak{su}(2) = \{A \in M_2(\mathbb{C}) : A^* = -A,\ \operatorname{tr}(A) = 0\}\) with basis \(\{i\sigma_j/2\}\) (Pauli matrices). The complexification is

\[ \mathfrak{su}(2)_\mathbb{C} \cong \mathfrak{sl}(2, \mathbb{C}) = \{A \in M_2(\mathbb{C}) : \operatorname{tr}(A) = 0\}, \]

with the standard \(\mathfrak{sl}(2)\) basis:

\[ H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad E = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad F = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}. \]

These satisfy the fundamental commutation relations:

\[ [H, E] = 2E, \quad [H, F] = -2F, \quad [E, F] = H. \]

Here \(H\) is the Cartan element (weight generator), \(E\) is the raising operator, and \(F\) is the lowering operator. The maximal torus of \(\mathrm{SU}(2)\) is \(T = \{\operatorname{diag}(e^{i\theta}, e^{-i\theta}) : \theta \in \mathbb{R}\}\) with Lie algebra \(\mathfrak{t} = \operatorname{span}_\mathbb{R}\{iH\} \subseteq \mathfrak{su}(2)\), and \(\mathfrak{t}_\mathbb{C} = \operatorname{span}_\mathbb{C}\{H\}\).

11.2 The Representations \(V_n\)

For each \(n \geq 0\), define

\[ V_n = \operatorname{span}_\mathbb{C}\{x^n, x^{n-1}y, x^{n-2}y^2, \ldots, y^n\} \subseteq \mathbb{C}[x, y], \]

the space of complex homogeneous polynomials of degree \(n\) in two variables. This is a vector space of dimension \(n+1\). The group \(\mathrm{SU}(2)\) acts on \(V_n\) by substitution: for

\[ A = \begin{pmatrix} a & b \\ -\bar{b} & \bar{a} \end{pmatrix} \in \mathrm{SU}(2), \]\[ (A \cdot f)(x, y) = f(\bar{a}x + \bar{b}y,\ -bx + ay). \]

(Here we substitute the columns of \(A^{-1} = \begin{pmatrix} \bar{a} & -b \\ \bar{b} & a \end{pmatrix}\) for \((x,y)\).)

Setting \(v_k = x^{n-k}y^k\) for \(k = 0, 1, \ldots, n\), the differentiated action \(\rho_*: \mathfrak{sl}(2,\mathbb{C}) \to \mathrm{End}(V_n)\) is:

Several key features are immediate. The element \(H\) acts diagonally on this basis, with eigenvalues \(n, n-2, \ldots, 2-n, -n\); these are the weights of \(V_n\). Each weight space is one-dimensional. The operator \(E\) raises the \(H\)-eigenvalue by \(2\) (it shifts \(v_k \mapsto v_{k-1}\)), and \(F\) lowers it by \(2\) (it shifts \(v_k \mapsto v_{k+1}\)).

11.3 Irreducibility of \(V_n\)

11.4 Complete Classification

11.5 The Clebsch-Gordan Formula

Having classified the irreducibles, the next natural question is: how does the tensor product of two irreducibles decompose?

11.6 Abstract \(\mathfrak{sl}(2,\mathbb{C})\) Theory and Applications

The theory developed here generalises in a fundamental way. Any semisimple Lie algebra \(\mathfrak{g}\) contains distinguished subalgebras isomorphic to \(\mathfrak{sl}(2,\mathbb{C})\), one for each root \(\alpha\): the \(\mathfrak{sl}(2)\)-triple \(\{H_\alpha, E_\alpha, F_{-\alpha}\}\) satisfying \([H_\alpha, E_\alpha] = 2E_\alpha\), \([H_\alpha, F_{-\alpha}] = -2F_{-\alpha}\), \([E_\alpha, F_{-\alpha}] = H_\alpha\). The representation theory of this subalgebra acting on any \(\mathfrak{g}\)-module gives a decomposition into \(V_n\)-summands. The integrality of weights (weights are integers for \(\mathfrak{sl}(2)\)-modules) propagates to the general setting. This \(\mathfrak{sl}(2)\)-trick is the heart of the proof that Cartan integers are integers (see Chapter 12) and of the Weyl character formula (Chapter 13).

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Chapter 12: Weights, Roots, and the Weyl Group

The representation theory of Chapter 11 relied on the structure of a single element \(H\) — the Cartan element — which simultaneously diagonalises all representations. For a general compact semisimple group \(G\), the role of \(H\) is played by the maximal torus \(T\). This chapter develops the theory of roots and weights, the combinatorial data encoded in the root system, and the Weyl group symmetry.

12.1 Complexification and Weight Decomposition

Fundamental examples: \(\mathfrak{su}(2)_\mathbb{C} \cong \mathfrak{sl}(2,\mathbb{C})\); \(\mathfrak{so}(n)_\mathbb{C} \cong \mathfrak{so}(n,\mathbb{C})\); \(\mathfrak{u}(n)_\mathbb{C} \cong \mathfrak{gl}(n,\mathbb{C})\); \(\mathfrak{su}(n)_\mathbb{C} \cong \mathfrak{sl}(n,\mathbb{C})\).

Fix a maximal torus \(T \subseteq G\) with Lie algebra \(\mathfrak{t} \subseteq \mathfrak{g}\). Every element of \(\mathfrak{t}\) generates a one-parameter subgroup of \(T\), and since \(T \cong (S^1)^\ell\) (where \(\ell = \dim T\) is the rank of \(G\)), these subgroups commute. In any representation \(\rho: G \to \mathrm{GL}(V)\), restricting \(\rho_*\) to \(\mathfrak{t}_\mathbb{C}\) gives a family of commuting diagonalisable operators, hence simultaneously diagonalisable.

The fundamental mechanism connecting roots and weights is:

12.2 Roots and Root Spaces

12.3 Root Systems

The pattern of roots carries enough information to classify all compact semisimple Lie groups. We abstract it into the notion of a root system.

The connection between root systems and Lie groups is: every compact simply connected semisimple Lie group is determined up to isomorphism by its root system, and every irreducible root system arises from a unique such group.

12.4 Positive Roots and Simple Roots

To work with root systems computationally, we choose an orientation.

The Cartan matrix of a root system is the \(\ell \times \ell\) integer matrix \(A_{ij} = \langle \alpha_i, \alpha_j^\vee \rangle = 2\langle \alpha_i, \alpha_j \rangle / \langle \alpha_j, \alpha_j \rangle\), where \(\alpha_1, \ldots, \alpha_\ell\) are the simple roots. It satisfies \(A_{ii} = 2\), \(A_{ij} \leq 0\) for \(i \neq j\), and \(A_{ij}A_{ji} \in \{0,1,2,3\}\). The Dynkin diagram encodes the Cartan matrix: nodes correspond to simple roots, and the number of edges between nodes \(i\) and \(j\) is \(A_{ij}A_{ji}\), with an arrow toward the shorter root when the lengths differ.

12.5 The Weyl Group

Key properties of \(W\):

• \(W\) is a finite group; \(|W|\) equals the number of Weyl chambers.
• \(W\) acts simply transitively on the set of Weyl chambers.
• For \(\mathrm{SU}(n+1)\): \(W \cong S_{n+1}\) (symmetric group), acting on \(\mathfrak{t}^*\) by permuting the coordinates \(\varepsilon_1, \ldots, \varepsilon_{n+1}\). Explicitly, the Weyl group of \(\mathrm{SU}(3)\) is \(S_3\), the dihedral group of order 6.
• For \(\mathrm{SO}(2n)\): \(W \cong S_n \ltimes (\mathbb{Z}/2)^{n-1}\), generated by permutations and even-cardinality sign changes of the coordinates.
• For \(\mathrm{SU}(2)\): \(W \cong S_2 = \{1, -1\}\), acting on \(\mathfrak{t}^* \cong \mathbb{R}\) by \(\pm 1\); concretely, the nontrivial element sends \(e^{i\theta} \mapsto e^{-i\theta}\) in \(T\).

The Weyl group plays a crucial role in the character theory of the next chapter: the Weyl character formula and the Weyl dimension formula both exhibit explicit \(W\)-symmetry.

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Chapter 13: Representations of Compact Semisimple Groups

With the machinery of roots, weights, and the Weyl group in place, we can now state and prove the crowning classification theorem of the course: every irreducible representation of a compact semisimple group is uniquely determined by its highest weight, and every dominant integral weight occurs.

13.1 Dominant Integral Weights

The weight lattice is the natural indexing set for representations.

The integrality condition \(\langle \lambda, \alpha^\vee \rangle \in \mathbb{Z}\) arises because any weight of a representation of the \(\mathfrak{sl}(2)\)-triple \(\{H_\alpha, E_\alpha, F_\alpha\}\) must be an integer (Theorem 11.3). The dominance condition \(\langle \lambda, \alpha^\vee \rangle \geq 0\) selects the “upper right” quadrant of the weight lattice — the intersection of the weight lattice with the closed dominant Weyl chamber.

The terminology is justified by Proposition 12.5: since \(E_\alpha\) shifts weights by \(\alpha > 0\), and \(E_\alpha \cdot v = 0\), there is no weight above \(\lambda\) in the representation.

13.2 The Theorem of the Highest Weight

13.3 The Weyl Character Formula

The Weyl character formula gives an explicit closed form for the character of any irreducible representation in terms of the root system and the highest weight.

The Weyl vector satisfies \(\langle \rho, \alpha_i^\vee \rangle = 1\) for all simple roots \(\alpha_i\), which is its most useful property in computations.

13.4 Worked Examples of the Weyl Formulas

13.5 The Topological Meaning of Integrality

The integrality condition for weights has a clean topological interpretation: a linear functional \(\lambda: \mathfrak{t} \to \mathbb{R}\) exponentiates to a well-defined character \(\chi_\lambda: T \to U(1)\), \(\chi_\lambda(\exp H) = e^{i\lambda(H)}\), if and only if \(\lambda\) lies in the weight lattice \(\Lambda\). The fundamental group \(\pi_1(G)\) is isomorphic (as an abelian group) to the quotient of the weight lattice by the root lattice \(Q = \mathbb{Z}\text{-span}(R)\):

\[ \pi_1(G) \cong \Lambda / Q. \]

Simply connected groups (like \(\mathrm{SU}(n)\)) have \(\Lambda = Q\) locally, while groups like \(\mathrm{U}(n)\) and \(\mathrm{SO}(n)\) have proper quotients. This connection between the representation theory and the topology of \(G\) is a deep and beautiful feature of the theory.

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Chapter 14: Further Topics

The preceding chapters have developed the full classification of irreducible representations of compact semisimple Lie groups. This final chapter brings together several threads: the Weyl integration formula (which connects character theory to torus integration), a synthesis of the main theorems, and an outlook toward the broader landscape of Lie theory.

14.1 The Weyl Integration Formula

The Weyl integration formula is an analogue of the change-of-variables formula for the conjugation map \(G/T \times T \to G\). It shows that, for the purpose of computing integrals of class functions, one only needs to integrate over the maximal torus — a dramatic reduction.

The Weyl integration formula has a fundamental application to the character inner product:

14.2 Tensor Product Decomposition

Given the classification theorem, a natural problem is to decompose tensor products of irreducible representations. This generalises the Clebsch-Gordan formula of Chapter 11.

For the classical groups in type \(A\), the multiplicities are computed by the Littlewood-Richardson rule (a combinatorial rule on Young tableaux). In general, the Klimyk formula expresses \(m(\lambda,\mu;\nu)\) in terms of weight multiplicities:

\[ m(\lambda,\mu;\nu) = \sum_{\mu' \in \mathrm{Wt}(V(\mu))} \dim V(\mu)_{\mu'} \cdot m(\lambda; \nu-\mu'), \]

where \(m(\lambda; \eta)\) is the multiplicity of the weight \(\eta\) in \(V(\lambda)\).

14.3 The Grand Synthesis

The following theorem collects the main results of the course into a single statement, capturing both the algebraic and analytic aspects of the theory.

This theorem is the culmination of a remarkable interplay between Lie theory, differential geometry, and functional analysis. The compact group \(G\) simultaneously encodes:

• Algebraic structure: the root system, Weyl group, and weight lattice are purely combinatorial objects classifying the representations.
• Geometric structure: the Haar measure, the maximal torus, and the conjugation map provide the geometric underpinning.
• Analytic structure: the \(L^2\) theory of the Peter-Weyl theorem connects representation theory to harmonic analysis.

14.4 Connections and Further Directions

We close with a brief orientation toward the broader landscape that opens beyond this course.

14.4.1 Structure of Simply Connected Simple Groups

The compact simply connected simple Lie groups are completely classified:

• Type \(A_n\) (\(n \geq 1\)): \(\mathrm{SU}(n+1)\), with Weyl group \(S_{n+1}\) and \(n\) simple roots.
• Type \(B_n\) (\(n \geq 2\)): \(\mathrm{Spin}(2n+1)\), the spin double-cover of \(\mathrm{SO}(2n+1)\).
• Type \(C_n\) (\(n \geq 3\)): \(\mathrm{Sp}(n)\), the compact symplectic group of \(n\times n\) quaternionic unitary matrices.
• Type \(D_n\) (\(n \geq 4\)): \(\mathrm{Spin}(2n)\), the spin double-cover of \(\mathrm{SO}(2n)\).
• Exceptional types: \(G_2\) (rank 2, dimension 14), \(F_4\) (rank 4, dimension 52), \(E_6\) (rank 6, dimension 78), \(E_7\) (rank 7, dimension 133), \(E_8\) (rank 8, dimension 248).

For a compact simple group \(G\), any other compact group with the same Lie algebra is a quotient \(G/Z\) for some subgroup \(Z \subseteq Z(G)\) of the centre; the centre satisfies \(Z(G) \cong \Lambda/Q\) (weight lattice modulo root lattice).

14.4.2 Real Forms and Non-Compact Groups

The complexification \(\mathfrak{g}_\mathbb{C}\) of a compact real Lie algebra \(\mathfrak{g}\) is a complex semisimple Lie algebra. The Dynkin diagram classification of complex semisimple Lie algebras exactly matches the classification of compact simple Lie groups: each Dynkin diagram corresponds to a unique complex simple Lie algebra and a unique compact real form.

However, a complex semisimple Lie algebra typically admits multiple non-isomorphic real forms — real Lie algebras whose complexification is the given complex algebra. For example:

• \(\mathfrak{su}(2)_\mathbb{C} \cong \mathfrak{sl}(2,\mathbb{C})\), which also has real form \(\mathfrak{sl}(2,\mathbb{R})\) (the Lie algebra of \(\mathrm{SL}(2,\mathbb{R})\)).
• \(\mathfrak{su}(n)_\mathbb{C} \cong \mathfrak{sl}(n,\mathbb{C})\), which also has real forms \(\mathfrak{sl}(n,\mathbb{R})\) and \(\mathfrak{su}(p,q)\) (the indefinite unitary algebras) for \(p+q=n\).

The non-compact real forms have a fundamentally different representation theory. The group \(\mathrm{SL}(2,\mathbb{R})\), for instance, has both finite-dimensional (non-unitary) and infinite-dimensional (unitary) irreducible representations. The classification of irreducible unitary representations of non-compact semisimple groups was achieved by Harish-Chandra and is an active area of research connected to the Langlands programme.

14.4.3 Infinite-Dimensional Representations

For a non-compact group like \(\mathrm{SL}(2,\mathbb{R})\), Bargmann’s classification (1947) identifies four families of unitary irreducibles:

• The discrete series: infinite-dimensional representations \(D_n^+\) and \(D_n^-\) (\(n \geq 1\)) whose matrix coefficients are square-integrable.
• The principal series: representations \(P_s\) (\(s \in i\mathbb{R}\)) induced from characters of the Borel subgroup.
• The complementary series: representations \(C_s\) (\(0 < s < 1\)) not unitarily induced.
• The trivial representation.

The general theory for semisimple groups, due to Harish-Chandra, Knapp-Zuckerman, Langlands, and Vogan, constructs irreducibles via cohomological induction and the geometry of flag varieties. The Langlands correspondence connects this representation theory to number theory via automorphic forms.

14.4.4 Algebraic Groups and Modular Representation Theory

Over fields of characteristic \(p > 0\), the Lie algebra no longer determines the representation theory of an algebraic group. New phenomena arise: the Frobenius endomorphism, restricted representations, and Steinberg’s tensor product theorem. The analogue of the Weyl character formula in characteristic \(p\) — the Lusztig conjecture (proved for large \(p\) by Andersen, Jantzen, and Soergel) — expresses the simple module characters in terms of Kazhdan-Lusztig polynomials, bringing geometric methods (perverse sheaves, intersection cohomology) into the heart of representation theory.

14.4.5 Geometric Representation Theory

The modern approach to representation theory — geometric representation theory — realises representations as cohomology groups of equivariant sheaves on algebraic varieties associated to \(G\) (flag varieties, nilpotent cones, Springer fibres). The Borel-Weil theorem gives a particularly clean realisation of the irreducibles \(V(\lambda)\) for compact groups:

This theorem — and its extension by Bott to higher cohomology groups — is the bridge between the algebraic representation theory developed in these notes and the modern geometric methods that dominate current research. The flag variety \(G_\mathbb{C}/B\), a projective algebraic variety, encodes the entire representation theory of \(G\) in its geometry, and the Weyl character formula arises from the Hirzebruch-Riemann-Roch theorem applied to \(\mathcal{L}_\lambda\). This perspective opens onto a vast and active research programme, connecting representation theory to algebraic geometry, \(D\)-modules, and mathematical physics.