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!

  • 1
    $\begingroup$ You should learn Lie algebras each day, but only post one question each day:) $\endgroup$ – Dietrich Burde Apr 22 at 15:45
  • $\begingroup$ My professor didn't think like you :) He want every student do Lie algebra expertly in one week. $\endgroup$ – 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.


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