Lie theory · Definition

Lie group

algebradifferential geometryLie theory

Before you start you need to know

Lie groups combine algebra and geometry: they are groups whose underlying set is also a smooth manifold, and whose operations respect that structure.

Definition

A Lie group \(G\) is a group that is also a smooth manifold. In this structure, multiplication and inversion are smooth functions.

\[\begin{gathered} \\ m:G\times G\to G,\quad (g,h)\mapsto g\cdot h \\ i:G\to G,\quad g\mapsto g^{-1} \\ \end{gathered}\]

If \(G\) is a Lie group, left translations \(L_g:h\mapsto gh\) are diffeomorphisms, so \(G\) looks the same near every point. The dimension of \(G\) as a manifold is an invariant of the group.

Guiding example: \((\mathbb{R},+)\) is a Lie group of dimension 1, since \(m(a,b)=a+b\) and \(i(a)=-a\) are smooth. The same holds for \(S^1=\{z\in\mathbb{C}:|z|=1\}\) with complex multiplication.

Typical error: thinking that being a group and a manifold separately is enough. Multiplication and inversion must be smooth with respect to the manifold structure; a topological group with an incompatible smooth structure is not a Lie group.

Think about it

Is it enough for \(G\) to be a group and a manifold separately to be a Lie group?

No: multiplication and inversion must be smooth functions with respect to the manifold structure.

Exercises 11

  1. What must a Lie group satisfy besides being a group?
  2. Besides multiplication, which other map must be smooth in a Lie group?
  3. What is the dimension of \((\mathbb{R},+)\) as a manifold?
  4. What is a smooth map between Lie groups that preserves the group operation called?
  5. Which of these sets with its operation is a Lie group?
  6. In \(S^1\), what is the inverse of \(e^{i\theta}\)?
  7. In \((\mathbb{R},+)\), what is the inverse of \(a\)?
  8. If \(G\) has dimension 3 and \(H\) is a closed Lie subgroup of dimension 1, what is the dimension of \(G/H\)?
  9. Which map is the left translation \(L_g\)?
  10. What is the identity element of \(S^1\) under complex multiplication?
  11. Which of these maps from \(\mathbb{R}\) to \(\mathbb{R}\) is a homomorphism from \((\mathbb{R},+)\) to \((\mathbb{R},+)\)?

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