1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Lie algebra: ideal

  1. Sep 22, 2011 #1
    1. The problem statement, all variables and given/known data

    Let [itex]\mathfrak{g}[/itex] be any lie algebra and [itex]\mathfrak{h}[/itex] be any ideal of [itex]\mathfrak{g}[/itex].

    The canonical homomorphism [itex]\pi : \mathfrak{g} \to \mathfrak{g/h}[/itex] is defined [itex]\pi (x) = x + \mathfrak{h}[/itex] for all [itex]x\in\mathfrak{g}[/itex].

    For any ideal [itex]\mathfrak{f}[/itex] of the quotient lie algebra [itex]\mathfrak{g/h}[/itex], consider the inverse image of [itex]\mathfrak{f}[/itex] in [itex]\mathfrak{g}[/itex] relative to [itex]\pi[/itex], that is: [tex]\pi ^{-1} (\mathfrak{f}) = \{X\in\mathfrak{g} : \pi (X)\in \mathfrak{f} \} .[/tex]
    Prove that [itex]\pi ^{-1} (\mathfrak{f})[/itex] is an ideal of the lie algebra [itex]\mathfrak{g}[/itex].

    3. The attempt at a solution

    See below
     
    Last edited: Sep 22, 2011
  2. jcsd
  3. Sep 22, 2011 #2
    This is my attempt:

    Let [itex]x\in\mathfrak{g}[/itex] and [itex]y\in\pi ^{-1}(\mathfrak{f})[/itex].

    We want to show that [itex][x,y]\in\pi ^{-1}(\mathfrak{f})[/itex]. That is, if [itex]\pi ([x,y]) \in \mathfrak{f}[/itex].

    Now [itex]\pi ([x,y]) = [\pi (x) , \pi (y)][/itex] since [itex]\pi[/itex] is a homomorphism.

    And since [itex]y\in \pi ^{-1}(\mathfrak{f})[/itex], [itex]\pi (y) \in \mathfrak{f}[/itex].

    But [itex][\pi (x) , \pi (y)] \in \mathfrak{f}[/itex] for all [itex]\pi (x) \in \mathfrak{g/h}[/itex] and [itex]\pi (y) \in \mathfrak{f}[/itex] since [itex]\mathfrak{f}[/itex] is an ideal of [itex]\mathfrak{g/h}[/itex] and [itex]\pi (x) = x+\mathfrak{h} \in \mathfrak{g/h}[/itex].

    Therefore [itex]\pi ([x,y])\in\mathfrak{f}[/itex] and [itex]\pi ^{-1}(\mathfrak{f})[/itex] is an ideal of [itex]\mathfrak{g}[/itex].

    Can anyone spot any mistakes or anything I've done wrong?
     
    Last edited: Sep 22, 2011
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Lie algebra: ideal
  1. Lie Algebras (Replies: 45)

  2. Derived lie algebra (Replies: 0)

  3. Help with Lie Algebra (Replies: 2)

Loading...