# Axioms of a homomorphism

1. Sep 24, 2011

### Ted123

1. The problem statement, all variables and given/known data

Let $\mathfrak{g}$ be a lie algebra over $\mathbb{C}$ and $\mathfrak{h}$ be an ideal of $\mathfrak{g}$.

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

3. The attempt at a solution

$\pi$ is linear:

For all $x,y\in\mathfrak{g}$ we have $$\pi (x+y) = (x+y)+\mathfrak{h} = (x+\mathfrak{h}) + (y+\mathfrak{h}) = \pi (x) + \pi (y)$$
If $\alpha\in\mathbb{C}$ then $$\pi (\alpha x) = \alpha x + \mathfrak{h} = \alpha (x + \mathfrak{h}) = \alpha \pi (x)$$
Furthermore, $$\pi ([x,y]) = [x,y] + \mathfrak{h} \stackrel{\stackrel{\mathfrak{h}\,\text{an ideal}}{\downarrow}}{=} [x,y] + [x,\mathfrak{h}] + [\mathfrak{h},y] + [\mathfrak{h},\mathfrak{h}] = [x+\mathfrak{h}, y+\mathfrak{h}] = [\pi (x),\pi (y)]$$
Are these all the axioms I need to check?

Last edited: Sep 24, 2011