Is \pi a Homomorphism of Lie Algebras for \mathfrak{g} and \mathfrak{h}?

[Ted123]

Homework Statement

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).

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?

 

[mighty2000]

Yes, these are all the axioms that you need to check to show that \pi is a homomorphism of lie algebras. You have correctly shown that \pi is linear, satisfies the scalar multiplication property, and preserves the lie bracket operation. This is enough to establish that \pi is a homomorphism of lie algebras. Great job on your solution!