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Ted123
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Homework Statement
• [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]
• [itex]\mathfrak{d}_3 \mathbb{C}[/itex] is the Lie algebra consisting of 3x3 diagonal matrices with complex entries with Lie bracket [itex][X,Y]=0[/itex] for all [itex]X,Y \in\mathfrak{d}_3 \mathbb{C}[/itex].
(We know [itex]\mathfrak{g}\cong \mathfrak{h}[/itex].)
Show that [itex]\mathfrak{d}_3 \mathbb{C}\ncong\mathfrak{g}[/itex] and [itex]\mathfrak{d}_3 \mathbb{C}\ncong\mathfrak{h}[/itex].
The Attempt at a Solution
A basis for [itex]\mathfrak{d}_3 \mathbb{C}[/itex] is [tex]\left\{ E=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} , F=\begin{bmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} , G=\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \right\}[/tex]
Is it sufficient to show that [itex][E,F]=0,\;[E,G]=0,\;[F,G]=0[/itex] which doesn't satisfy all the lie bracket relations in [itex]\mathfrak{g}[/itex] so [itex]\mathfrak{d}_3 \mathbb{C}\ncong\mathfrak{g}[/itex]?
And since [itex]\mathfrak{d}_3 \mathbb{C}\ncong\mathfrak{g}\cong \mathfrak{h},\;\mathfrak{d}_3 \mathbb{C}\ncong\mathfrak{h}[/itex] but to show it:
[itex]X=\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} , Y=\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}\in \mathfrak{h}[/itex]
but [itex][X,Y]=XY-YX=\begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \neq \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}[/itex]
so 2 matrices in [itex]\mathfrak{h}[/itex] don't satisfy the Lie bracket in [itex]\mathfrak{d}_3 \mathbb{C}[/itex] so [itex]\mathfrak{d}_3 \mathbb{C}\ncong\mathfrak{h}[/itex]?
So is all I need to do to show 2 lie algebras are not isomorphic is to provide a counterexample of how matrices in 1 lie algebra don't satisfy the lie bracket in another lie algebra; therefore the 2 lie algebras can't be isomorphic as isomorphisms preserve lie brackets?
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