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Lie algebra dimension

  1. Sep 16, 2011 #1
    1. The problem statement, all variables and given/known data

    [itex]\mathfrak{g}[/itex] is the Lie algebra with basis vectors [itex]E,F,G[/itex] such that the following relations for Lie brackets are satisfied:

    [itex][E,F]=G,\;\;[E,G]=0,\;\;[F,G]=0.[/itex]

    [itex]\mathfrak{h}[/itex] is the Lie algebra consisting of 3x3 matrices of the form

    [itex]\begin{bmatrix} 0 & a & c \\ 0 & 0 & b \\ 0 & 0 & 0 \end{bmatrix}[/itex] where [itex]a,b,c[/itex] are any complex numbers. The vector addition and scalar multiplication on [itex]\mathfrak{h}[/itex] are the usual operations on matrices.

    The Lie bracket on [itex]\mathfrak{h}[/itex] is defined as the matrix commutator: [itex][X,Y] = XY - YX[/itex] for any [itex]X,Y \in \mathfrak{h}.[/itex]

    3. The attempt at a solution

    Is the dimension of [itex]\mathfrak{g}[/itex] and [itex]\mathfrak{h}[/itex] both 3?
     
  2. jcsd
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