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:
  • the general linear group \(\mathrm{GL}(n, \mathbb{C})\)
  • the general linear group \(\mathrm{GL}(n, \mathbb{R})\)
  • the special linear group \(\mathrm{SL}(n, \mathbb{C})\)
  • the special linear group \(\mathrm{SL}(n, \mathbb{R})\)
  • the unitary group \(\mathrm{U}(n)\)
  • the orthogonal group \(\mathrm{O}(n)\)
  • the special unitary group \(\mathrm{SU}(n)\)
  • the special orthogonal group \(\mathrm{SO}(n)\)
  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!

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

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