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Axioms of a homomorphism

  1. Sep 24, 2011 #1
    1. The problem statement, all variables and given/known data

    Let [itex]\mathfrak{g}[/itex] be a lie algebra over [itex]\mathbb{C}[/itex] and [itex]\mathfrak{h}[/itex] be an ideal of [itex]\mathfrak{g}[/itex].

    Show that the map [itex]\pi : \mathfrak{g} \to \mathfrak{g/h}[/itex] defined by [itex]\pi (x) = x + \mathfrak{h}[/itex] for all [itex]x\in\mathfrak{g}[/itex] satisfies all the axioms of a homomorphism of lie algebras (called the canonical homomorphism).

    3. The attempt at a solution

    [itex]\pi[/itex] is linear:

    For all [itex]x,y\in\mathfrak{g}[/itex] we have [tex]\pi (x+y) = (x+y)+\mathfrak{h} = (x+\mathfrak{h}) + (y+\mathfrak{h}) = \pi (x) + \pi (y)[/tex]
    If [itex]\alpha\in\mathbb{C}[/itex] then [tex]\pi (\alpha x) = \alpha x + \mathfrak{h} = \alpha (x + \mathfrak{h}) = \alpha \pi (x)[/tex]
    Furthermore, [tex]\pi ([x,y]) = [x,y] + \mathfrak{h} \stackrel{\stackrel{\mathfrak{h}\,\text{an ideal}}{\downarrow}}{=} [x,y] + [x,\mathfrak{h}] + [\mathfrak{h},y] + [\mathfrak{h},\mathfrak{h}] = [x+\mathfrak{h}, y+\mathfrak{h}] = [\pi (x),\pi (y)][/tex]
    Are these all the axioms I need to check?
    Last edited: Sep 24, 2011
  2. jcsd
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