Exercise · ZW-LIE-GROUP-011 · ★★★ Advanced
Which of these maps from \(\mathbb{R}\) to \(\mathbb{R}\) is a homomorphism from \((\mathbb{R},+)\) to \((\mathbb{R},+)\)?
Choose one option.
Show answer
- D. \(x\mapsto 2x\)
Explanation
A homomorphism must satisfy \(f(a+b)=f(a)+f(b)\). For \(2x\) we get \(2(a+b)=2a+2b\), which holds; \(x^2\), \(x+1\) and \(e^x\) fail with \(a=b=1\).
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-005 Which of these sets with its operation is a Lie group?
- 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?