Exercise · ZW-LIE-GROUP-005 · ★★☆ Intermediate
Which of these sets with its operation is a Lie group?
Choose one option.
Show answer
- B. \(S^1\) with complex multiplication
Explanation
\(S^1\) is a 1-dimensional manifold with smooth product and inverse. \((\mathbb{Q},+)\) is not a manifold, and \([0,\infty)\) has no inverses.
Review the concept: Lie group
A Lie group \(G\) is a group that is also a smooth manifold. In this structure, multiplication and inversion are smooth functions.
\[\begin{gathered} \\ m:G\times G\to G,\quad (g,h)\mapsto g\cdot h \\ i:G\to G,\quad g\mapsto g^{-1} \\ \end{gathered}\]
More practice problems: lie group
- ZW-LIE-GROUP-001 What must a Lie group satisfy besides being a group?
- ZW-LIE-GROUP-002 Besides multiplication, which other map must be smooth in a Lie group?
- ZW-LIE-GROUP-003 What is the dimension of (ℝ,+) as a manifold?
- ZW-LIE-GROUP-004 What is a smooth map between Lie groups that preserves the group operation called?
- ZW-LIE-GROUP-006 In S^1, what is the inverse of e^iθ?
- ZW-LIE-GROUP-007 In (ℝ,+), what is the inverse of a?
- ZW-LIE-GROUP-008 If G has dimension 3 and H is a closed Lie subgroup of dimension 1, what is the dimension of G/H?
- ZW-LIE-GROUP-009 Which map is the left translation L_g?