### Lie Groups

#### John Baez

• Part 1: introduction, definition of smooth manifolds, smooth maps, products of manifolds, Lie groups, the general linear groups $$\mathrm{GL}(n,\mathbb{R})$$ and $$\mathrm{GL}(n,\mathbb{C})$$, the closed subgroup theorem, and the definition of matrix Lie groups.
• Part 2: Lie group homomorphisms, the category of Lie groups. Examples of matrix Lie groups: Picturing all these groups in the case $$n = 1$$.
• Part 3: Picturing some of these groups in the case $$n = 2$$. Why $$\mathrm{SO}(2) \cong \mathrm{U}(1) \cong S^1$$. Why $$\mathrm{O}(n)$$ is diffeomorphic to the disjoint union of two copies of $$\mathrm{SO}(n)$$. Why $$\mathrm{SL}(2,\mathbb{R})$$ is diffeomorphic to $$\mathbb{R}^2 \times S^1$$, and how $$\mathrm{SO}(2)$$ sits inside $$\mathrm{SL}(2,\mathbb{R})$$. Examples of maximal compact subgroups. Every connected Lie group $$G$$ is diffeomorphic to $$K \times \mathbb{R}^n$$ for a maximal compact subgroup $$K \subseteq G$$, which is unique up to conjugation.
• Part 4: The three associative normed division algebra: the real numbers $$\mathbb{R}$$, the complex numbers $$\mathbb{C}$$ and the quaternions $$\mathbb{H}$$. Defining the quaternions and proving that they form a normed division algebra. Proof that as Lie groups, $$\mathrm{SU}(2)$$ is isomorphic to $$\mathrm{Sp}(1) = \{a \in \mathbb{H}: \; |a| = 1 \}.$$ Quaternionic matrix groups: the quaternionic general linear group: $$\mathrm{GL}(n,\mathbb{H}) = \{g \in \mathrm{M}_n(\mathbb{H}) : \; g \textrm{ invertible} \}$$ the quaternionic special linear group: $$\mathrm{SL}(n,\mathbb{H}) = \{g \in \mathrm{M}_n(\mathbb{H}) : \; \det(g) = 1 \}$$ and the quaternionic unitary group, also called the compact symplectic group: $$\mathrm{Sp}(n) = \{g \in \mathrm{M}_n(\mathbb{H}) : \; gg^\ast = 1 \}$$ A warning about quaternionic determinants: they take values in the complex numbers, and $$gg^\ast = 1$$ implies $$\mathrm{det}(g) = 1$$ so $$\mathrm{Sp}(n,\mathbb{H}) \subseteq \mathrm{SL}(n,\mathbb{H})$$. The three Lie groups that are spheres: $$S^0 \cong \{x \in \mathbb{R}: \; |x| = 1\} = \mathrm{O}(1) \cong \mathbb{Z}_2$$ $$S^1 \cong \{x \in \mathbb{C}: \; |x| = 1\} = \mathrm{U}(1) \cong \mathrm{O}(2)$$ $$S^3 \cong \{x \in \mathbb{H}: \; |x| = 1\} = \mathrm{Sp}(1) \cong \mathrm{SU}(2)$$
• Part 5: Covers of Lie groups, and universal covers. The group $$\mathbb{R}$$ is a universal cover of $$\mathrm{SO}(2)$$. The spin group $$\mathrm{Spin}(n)$$ is, by definition, the universal cover of $$\mathrm{SO}(n)$$ for $$n \ge 2$$. Examples:

• Spin(3) $$\cong \mathrm{SU}(2) \cong \mathrm{Sp}(1)$$
• Spin(4) $$\cong \mathrm{SU}(2) \times \mathrm{SU}(2)$$
• Spin(5) $$\cong \mathrm{Sp}(2)$$
• Spin(6) $$\cong \mathrm{SU}(4)$$

Proof that $$\mathrm{SO}(3)$$ is diffeomorphic to the projective space $$\mathbb{R}\mathrm{P}^3$$, which has fundamental group $$\mathbb{Z}_2$$. Proof that the universal cover of $$\mathrm{SO}(3)$$ is $$\mathrm{SU}(2)$$. Picture proof that the fundamental group of $$\mathrm{SO}(3)$$ is $$\mathbb{Z}_2$$: this is a fancy version of the belt trick.
Note that after the cube rotates 720°, the belts are back to their original positions!

• Part 6: Compact Lie groups. The Lie groups $$\mathrm{O}(n), \mathrm{SO}(n), \mathrm{Spin}(n), \mathrm{U}(n), \mathrm{SU}(n)$$ and $$\mathrm{Sp}(n)$$ are compact. The unfeasibility of classifying all compact Lie groups. The classification of compact connected Lie groups. Every compact connected Lie group has a finite cover that is a product of a compact connected abelian Lie group and a compact simply connected Lie group. Every compact connected abelian Lie group is isomorphic to a torus $$\mathbb{T}^n = \mathrm{U}(1) \times \cdots \times \mathrm{U}(1).$$ Every compact simply connected Lie group is isomorphic to a product of finitely many copies of the classical compact Lie groups $$\mathrm{Spin}(n), \mathrm{SU}(n), \mathrm{Sp}(n)$$ and the five exceptional compact Lie groups, built using octonions:

• G2, the automorphism group of the octonions, which is 14-dimensional.
• F4, the isometry group of the octonionic projective plane, which is 52-dimensional.
• E6, the isometry group of the bioctonionic projective plane, which is 78-dimensional.
• E7, the isometry group of the quateroctonionic projective plane, which is 133-dimensional.
• E8, the isometry group of the octooctonionic projective plane, which is 248-dimensional: the king of Lie groups!

Example: $$\mathrm{U}(n)$$ has $$\mathrm{U}(1) \times \mathrm{SU}(n)$$ as an $$n$$-fold cover.

• Part 7: Lie group actions. Transporting an action along diffeomorphisms. Restricting an action to a closed subgroup.
• Part 8: Felix Klein's idea for reducing geometry to group theory: his Erlangen program. Euclidean geometry and the Euclidean group. The product of actions; restricting an action to an invariant submanifold. Transitive actions. How a Lie group gives a geometry and an action gives a type of figure. Real projective plane geometry and the group $$\mathrm{GL}(3,\mathbb{R})$$: points, lines and point-line flags. Higher-dimensional real projective geometry and the group $$\mathrm{GL}(n,\mathbb{R})$$: Grassmannians and flag manifolds. Complex projective geometry and $$\mathrm{GL}(n, \mathbb{C})$$, quaternionic projective geometry and $$\mathrm{GL}(n,\mathbb{H})$$. For more, see:

• Part 9: Lie group representations. A representation of a Lie group $$G$$ on a finite-dimensional vector space $$V$$ amounts to the same thing as a Lie group homomorphism $$\rho \colon G \to \mathrm{GL}(V)$$ where the general linear group $$\mathrm{GL}(V)$$ is the Lie group of all invertible linear transformations of $$V$$. For any Lie group $$G$$ there is a category $$\mathrm{Act}(G)$$ of actions of $$G$$ and a subcategory $$\mathrm{Rep}(G)$$ of representations of $$G$$. Example: the tautological representations of $$\mathrm{GL}(n,\mathbb{R}), \mathrm{GL}(n,\mathbb{C})$$ and closed subgroups of these groups. The direct sum and tensor product of representations. Proof that given a representation $$\rho \colon G \to \mathrm{GL}(V)$$ we have $$\rho \otimes \rho \; \cong \; S^2 V \, \oplus \Lambda^2 V$$ where $$S^2 V \subseteq V \otimes V$$ is the space of symmetric tensors and $$\Lambda^2 V \subseteq V \otimes V$$ is the space of antisymmetric tensors. The example of this where $$\rho$$ is the tautological representation of $$\mathrm{GL}(2,\mathbb{C})$$ on $$\mathbb{C}^2$$. Definition of invariant subspace and subrepresentation. A tiny taste of Young diagrams.
• Part 10: Lie groups in quantum physics. Hilbert spaces. In quantum physics any system is described using some Hilbert space $$H$$, and the states of that system are described by unit vectors $$\psi \in H$$, that is, vectors $$\psi$$ with $$\|\psi\| = 1$$. A linear operator $$U \colon H \to H'$$ that maps unit vectors to unit vector is called an isometry, and this is the same as a linear operator that preserves the norm, or the inner product. An invertible isometry is called a unitary operator. A linear operator $$U \colon H \to H$$ is unitary iff $$UU^\ast = U^\ast U = 1_H$$, and for any a Hilbert space $$H$$ the unitary operators $$\mathrm{U}(H) = \{ U \colon H \to H : \; U \textrm{ is unitary} \}$$ form a Lie group. Any finite-dimensional Hilbert space $$H$$ is isomorphic to $$\mathbb{C}^n$$ with its standard inner product, and then $$U(H) \cong \mathrm{U}(n)$$ is a Lie group. A unitary representation of a Lie group $$G$$ on a finite-dimensional Hilbert space $$H$$ is a Lie group homomorphism $$\rho \colon G \to \mathrm{U}(H)$$. Examples: the spin-0 particle, the spin-1/2 particle, and the spin-1 particle are unitary representations $$\alpha_0 \colon \mathrm{SU}(2) \to \mathrm{U}(\mathbb{C}^1) \cong \mathrm{U}(1)$$ $$\alpha_{1/2} \colon \mathrm{SU}(2) \to \mathrm{U}(\mathbb{C}^2) \cong \mathrm{U}(2)$$ $$\alpha_1 \colon \mathrm{SU}(2) \to \mathrm{U}(\mathbb{C}^3) \cong \mathrm{U}(3)$$ Rotating a spin-1 particle. How tensor products of representations describe ways of 'sticking together' physical systems to build new physical system. When we stick together two spin-1/2 particles we can get either a spin-0 particle or a spin-1 particles, due to the isomorphism of representations $$\alpha_{1/2} \otimes \alpha_{1/2} \cong \alpha_0 \oplus \alpha_1$$ which arises from the isomorphism $$\rho \otimes \rho \; \cong \; \Lambda^2 V \, \oplus S^2 V.$$
• Part 11: Lie algebras: a quick overview of some essential facts which we will study in more detail later. The Lie algebra $$\mathfrak{g}$$ of a Lie group $$G$$ is its tangent space at the identity. There's a one-to-one correspondence between elements $$v \in \mathfrak{g}$$ and Lie group homomorphisms $$C \colon \mathbb{R} \to G$$, given by the requirement that $$C'(0) = v$$. The Lie algebra $$\mathfrak{g}$$ acquires, in a manner to be explained later, an operation $$[\cdot,\cdot] : \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}$$ that is bilinear, antisymmetric and obeys the Jacobi identity. Conversely, any finite-dimensional vector space with an operation with these 3 properties arises (up to isomorphism) by taking the tangent space of some Lie group $$G$$. Furthermore, this Lie group is unique (up to isomorphism) if we require that it's connected and simply connected.

• Part 12: The exponential map. For any Lie group $$G$$ and any $$v \in \mathfrak{g}$$ there exists a unique Lie group homomorphism $$C \colon \mathbb{R} \to G$$ such that $$C'(0) = v$$. We prove this for matrix Lie groups using the matrix exponential: the exponential of an $$n \times n$$ complex matrix $$A \in M_n(\mathbb{C})$$ is defined by $$\exp(A) = \sum_{n = 0}^\infty \frac{A^n}{n!}$$ and we get the Lie group homomorphism $$C \colon \mathbb{R} \to G$$ by setting $$C(t) = \exp(t v).$$ Our proof also uses the basic theorem on the local existence and uniqueness of solutions of first-order ordinary differential equations.

• Part 13: The Lie bracket. How the Baker–Campbell–Hausdorff formula lets us recover the product in a matrix Lie group, at least near the identity, from the Lie bracket. Proof that if $$\mathfrak{g}$$ is the Lie algebra of matrix Lie group $$G$$ then $$v, w \in \mathfrak{g} \implies vw - wv \in \mathfrak{g}.$$ In this case we can start with the smooth map $$\begin{array}{rccl} \mathrm{AD}(g) \colon & G & \to & G \\ & h & \mapsto & ghg^{-1} \end{array}$$ for each $$g\in G$$, then differentiate it to get $$\begin{array}{rccl} \mathrm{Ad}(g) \colon & \mathfrak{g} & \to & \mathfrak{g} \\ & w & \mapsto & gwg^{-1} \end{array}$$ and then differentate this with respect to $$g$$ to get a linear map $$\begin{array}{rccl} \mathrm{ad}(v) \colon & \mathfrak{g} & \to & \mathfrak{g} \\ & w & \mapsto & vw - wv \end{array}$$ for each $$v \in \mathfrak{g}$$. Thus, in this case we have an operation $$[\cdot, \cdot] \colon \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}$$ given by $$[v,w] = vw - wv .$$ We define the Lie bracket for a general Lie group by copying this procedure: we start with $$\begin{array}{rccl} \mathrm{AD}(g) \colon & G & \to & G \\ & h & \mapsto & ghg^{-1} , \end{array}$$ then we differentiate it to get $$\mathrm{Ad}(g) \colon \mathfrak{g} \to \mathfrak{g}$$ and $$\mathrm{ad}(v) \colon \mathfrak{g} \to \mathfrak{g},$$ and then define $$[v,w] = \mathrm{ad}(v)(w) .$$ Proof that the Lie bracket is linear in each argument, antisymmetric and obeys the Jacobi identity.

• Part 14: Matrix Lie algebras, also called classical Lie algebras. The Lie algebras of $$\mathrm{GL}(n,\mathbb{C}), \mathrm{GL}(n, \mathbb{R})$$ and $$\mathrm{GL}(n,\mathbb{H})$$ are $$\mathfrak{gl}(n,\mathbb{C}) = \mathrm{M}_n(\mathbb{C})$$ $$\mathfrak{gl}(n,\mathbb{R}) = \mathrm{M}_n(\mathbb{R})$$ $$\mathfrak{gl}(n,\mathbb{H}) = \mathrm{M}_n(\mathbb{H})$$ The Lie algebras of $$\mathrm{SL}(n,\mathbb{C}), \mathrm{SL}(n, \mathbb{R})$$ and $$\mathrm{SL}(n,\mathbb{H})$$ are $$\mathfrak{gl}(n,\mathbb{C}) = \{a \in \mathrm{M}_n(\mathbb{C}): \mathrm{tr} (a) = 0\}$$ $$\mathfrak{gl}(n,\mathbb{R}) = \{a \in \mathrm{M}_n(\mathbb{R}): \mathrm{tr} (a) = 0\}$$ $$\mathfrak{gl}(n,\mathbb{H}) = \{a \in \mathrm{M}_n(\mathbb{H}): \mathrm{tr} (a) = 0\}$$ where in the last case we define a complex-valued trace of an $$n \times n$$ quaternionic matrix by regarding it as a $$2n \times 2n$$ complex matrix. The Lie algebras of $$\mathrm{U}(n), \mathrm{O}(n)$$ and $$\mathrm{Sp}(n)$$ are $$\mathfrak{u}(n) = \{a \in \mathrm{M}_n(\mathbb{C}) : a + a^\ast = 0 \}$$ $$\mathfrak{o}(n) = \{a \in \mathrm{M}_n(\mathbb{R}) : a + a^\ast = 0 \}$$ $$\mathfrak{sp}(n) = \{a \in \mathrm{M}_n(\mathbb{H}) : a + a^\ast = 0 \}$$ A matrix with $$a + a^\ast = 0$$ is called skew-adjoint. Finally, the Lie algebras of $$\mathrm{SU}(n)$$ and $$\mathrm{SO}(n)$$ are $$\mathfrak{su}(n) = \{a \in \mathrm{M}_n(\mathbb{C}) : \mathrm{tr}(a) = 0, a + a^\ast = 0 \}$$ $$\mathfrak{so}(n) = \mathfrak{o}(n) = \{a \in \mathrm{M}_n(\mathbb{H}) : a + a^\ast = 0 \}$$ where the final surprise happens because $$a + a^\ast = 0$$ implies $$\mathrm{tr}(a) = 0$$ for real matrices.

• Part 15: From Lie groups to Lie algebras. Every Lie group homomorphism gives a homomorphism of Lie algebras. This defines a functor from the category of Lie groups to Lie algebras, so our big commuting diagram of Lie groups from Part 4 gives a big commuting diagram of Lie algebras.

• Part 16: Lie algebra representations. How a Lie group representation $$\rho\colon G \to GL(V)$$ gives a Lie algebra representation $$d\rho \colon \mathfrak{g} \to \mathfrak{gl}(V)$$, and a unitary Lie group representation $$\rho\colon G \to U(H)$$ gives a Lie algebra representation $$d\rho \colon \mathfrak{g} \to \mathfrak{u}(H)$$.

• Part 17: Lie algebras in quantum mechanics. In quantum physics, self-adjoint operators on the Hilbert space of a system correspond to observables. Given a unitary representation $$\rho \colon G \to U(H)$$ of a Lie group on a finite-dimensional Hilbert space, we get an observable $$id\rho(v) \colon H \to H$$ for each element $$v \in \mathfrak{g}$$. How to compute the probability that we get a specific answer when we measure some observable in some state. Example: the $$z$$ component of angular momentum for the spin-1/2 particle.

• Part 18: A many-faceted Lie algebra. The following Lie algebras are all isomorphic:

• $$\mathfrak{o}(3) = \{a \in \mathrm{M}_3(\mathbb{R}) : \, a^\ast = -a \}$$
• $$\mathfrak{so}(3) = \{a \in \mathrm{M}_3(\mathbb{R}) : \, a^\ast = -a, \mathrm{tr}(a) = 0 \}$$
• $$\mathfrak{su}(2) = \{a \in \mathrm{M}_2(\mathbb{C}) : \, a^\ast = -a, \mathrm{tr}(a) = 0 \}$$
• $$\mathfrak{sp}(1) = \{a \in \mathbb{H} : \, a^\ast = -a \}$$
• $$\mathbb{R}^3$$ with its cross product as Lie bracket.

The isomorphism $$\mathfrak{sp}(1) \cong \mathfrak{su}(2)$$ maps the imaginary quaternions $$i,j,k$$ to skew-adjoint matrices $$I = \left( \begin{array}{cc} 0 & -i \\ -i & 0 \end{array} \right) ,\quad J = \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right) , \quad K = \left( \begin{array}{cc} -i & 0 \\ 0 & -i \end{array} \right).$$ Physicists turn these into self-adjoint matrices by multiplying them by $$i$$, and call the resulting self-adjoint matrices the Pauli matrices $$\sigma_1, \sigma_2, \sigma_3$$.

• Part 19 - The spin-1 particle. Why the angular momentum of a spin-1 particle along the $$z$$ axis can only equal $$1, 0$$ or $$-1$$. The spin-1/2 particle revisited: why the angular momentum of a spin-1/2 particle along the $$z$$ axis can only equal $$+\frac{1}{2}$$ or $$-\frac{1}{2}$$.
• Part 20 - Irreducible representations of Lie groups. Invariant subspaces, subrepresentations, reducible and decomposable representations. In what follows we consider only finite-dimensional representations of Lie groups.

Every decomposable representation is reducible, but not conversely: there is a 2-dimensional representation of the Lie group $$\mathbb{R}$$ that is reducible but not decomposable. However, every unitary representation that is reducible is decomposable. Every representation of a Lie group is a direct sum of finitely many indecomposable representations. Every representation of a compact Lie group can be given an inner product making it unitary. Putting these results together, we see that every representation of a compact Lie group is the direct sum of finitely many irreducible representations.

• Part 21: Irreducible representations of SU(2). There is a representation of $$\mathrm{SU}(2)$$ on $$S^n \mathbb{C}^2$$, which is the space of homogeneous polynomials of degree $$n$$ in two variables. If we set $$j = n/2$$, physicists call this the spin-$$j$$ representation $$\alpha_j \colon \mathrm{SU}(2) \to \mathrm{GL}(S^n \mathbb{C}^2)$$ and it generalizes the cases we've already seen, where $$j = 0,\frac{1}{2}$$ and $$1$$. We construct the spin-$$j$$ representation and prove that it is irreducible. The proof involves first building a representation $$\rho \colon \mathrm{GL}(2,\mathbb{C}) \to \mathrm{GL}(S^n \mathbb{C}^2),$$ and then restricting it to get the representation $$\alpha_j$$ of $$\mathrm{SU}(2)$$. To prove this is irreducible it helps to show any representation of $$\mathfrak{su}(n)$$ on a complex vector space extends to a representation of $$\mathfrak{sl}(n,\mathbb{C}$$. In particular $$d\alpha_j \colon \mathfrak{su}(2) \to \mathfrak{gl}(S^n \mathbb{C}^2)$$ extends to a representation $$d\tilde{\alpha}_j \colon \mathfrak{sl}(2,\mathbb{C}) \to \mathfrak{gl}(S^n \mathbb{C}^2)$$ which is a restriction of $$d\rho \colon \mathfrak{gl}(2,\mathbb{C}) \to \mathfrak{gl}(S^n \mathbb{C}^2)$$ This lets us use raising and lowering operators on $$S^n \mathbb{C}^2$$, also called ladder operators. For other proofs, see:

• Part 22: From $$\mathrm{SU}(2)$$ to $$\mathrm{SU}(3)$$ and beyond. Isospin and representations of $$\mathrm{SU}(2)$$. Gell-Mann's theory of quarks and representations of SU(3).

• Part 23: Lie groups rule the world! A reminder of how symmetries give observables. Time translation gives energy; spatial translations give momentum; rotations give angular momentum. Spatial translations and rotations can be combined into the 6-dimensional Euclidean group, which has an action on space, $$\mathbb{R}^3$$. Time translations and the Euclidean group can be combined into a 7-dimensional Lie group that has an action on spacetime, $$\mathbb{R}^4$$. This group is a subgroup of an even larger group that acts on spacetime, the 10-dimensional Galilei group. The Galilei group also includes symmetries called boosts, which give another important observable: position. Einstein's special relativity replaces the Galilei group with another 10-dimensional Lie group acting on spacetime: the Poincaré group. In the Standard Model of particle physics, the symmetry group is the product of the Poincaré group and $$\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1).$$ Here $$\mathrm{SU}(3)$$ is the symmetry group of the strong force, $$\mathrm{SU}(2)$$ is the symmetry group of the weak force and $$\mathrm{U}(1)$$ is the symmetry group of the electromagnetic force (roughly).

Reality favors symmetry — Jorge Luis Borges The root vectors of $$\mathrm{Spin}(8)$$, the double cover of the rotation group in 8 dimensions.

© 2021 John Baez
baez@math.removethis.ucr.andthis.edu