### 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.
• 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. 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!

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