# Homework Help: If V is a 3-dimensional Lie algebra with basis vectors E,F,G

1. Jun 9, 2012

### Ted123

If V is a 3-dimensional Lie algebra with basis vectors E,F,G with Lie bracket relations [E,F]=G, [E,G]=0, [F,G]=0 and V' is the Lie algebra consisting of all 3x3 strictly upper triangular matrices with complex entries then would you say the following 2 mappings (isomorphisms) are different? I had to give an example of 2 different isomorphisms between these vector spaces.

$\varphi : V \to V'$ given by

$\varphi(aE+bF+cG)=\left( \begin{array}{ccc} 0 & a & c\\ 0 & 0 & b\\ 0 & 0 & 0 \end{array} \right)\;,\;\;\;\;\;a,b,c \in \mathbb{C}$

and $\bar{\varphi} : V \to V'$ given by

$\bar{\varphi}(E)=\left( \begin{array}{ccc} 0 & 1 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array} \right)$

$\bar{\varphi}(F)=\left( \begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end{array} \right)$

$\bar{\varphi}(G)=\left( \begin{array}{ccc} 0 & 0 & 1\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array} \right)$

2. Jun 9, 2012

### algebrat

Re: Mappings

Those are the same. Can you switch E and F? Is that different? Anyone, Bueller?