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

  1. Jun 9, 2012 #1
    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.

    [itex]\varphi : V \to V'[/itex] given by

    [itex]\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}[/itex]

    and [itex]\bar{\varphi} : V \to V'[/itex] given by

    [itex]\bar{\varphi}(E)=\left( \begin{array}{ccc} 0 & 1 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array} \right)[/itex]

    [itex]\bar{\varphi}(F)=\left( \begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end{array} \right)[/itex]

    [itex]\bar{\varphi}(G)=\left( \begin{array}{ccc} 0 & 0 & 1\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array} \right)[/itex]
     
  2. jcsd
  3. Jun 9, 2012 #2
    Re: Mappings

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