# Show that $\mathfrak{b}_3 (\mathbb{C}) / \mathfrak{n}_3 (\mathbb{C}) \cong \mathfrak{t}_3 (\mathbb{C})$

Show that $$\mathfrak{b}_3 (\mathbb{C}) / \mathfrak{n}_3 (\mathbb{C}) \cong \mathfrak{t}_3 (\mathbb{C})$$, in which $$\mathfrak{b}_3 (\mathbb{C}),\mathfrak{n}_3 (\mathbb{C}),\mathfrak{t}_3 (\mathbb{C})$$ are space all upper triangular matrices, space all strictly upper triangular matrices, diagonal matrices $$3 \times 3$$ respectively.

How to do this? I just "learn" Lie algebra for two days. Thank all!

• You should learn Lie algebras each day, but only post one question each day:) – Dietrich Burde Apr 22 at 15:45
• My professor didn't think like you :) He want every student do Lie algebra expertly in one week. – Minh Apr 22 at 15:48

Consider the map $$\phi:\mathfrak{b}_3(\Bbb C)\to\mathfrak{t}_3(\Bbb C)$$ which takes an upper triangular matrix $$M$$ to the diagonal matrix with the same diagonal entries. (i) Prove that $$\phi$$ is a homomorphism of associative algebras. (ii) Prove that $$\phi$$ is a homomorphism of Lie algebras. (iii) Prove its kernel is $$\mathfrak{n}_3(\Bbb C)$$. (iv) Apply the First Homomorphism Theorem for Lie algebras.
This works for other values of $$3$$ too.