- #1
Ted123
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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)
\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)