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Ted123
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Homework Statement
Let [tex]\mathfrak{g} , \mathfrak{h}[/tex] be Lie algebras over [tex]\mathbb{C}.[/tex]
(i) When is a mapping [tex]\varphi : \mathfrak{g} \to \mathfrak{h}[/tex] a homomorphism?
(ii) When are the Lie algebras [tex]\mathfrak{g}[/tex] and [tex]\mathfrak{h}[/tex] isomorphic?
(iii) Let [tex]\mathfrak{g}[/tex] be the Lie algebra with basis vectors E,F,G such that the following relations for Lie brackets are satisfied: [tex][E,F]=G,\;\;[E,G]=0,\;\;[F,G]=0.[/tex] Let [tex]\mathfrak{h}[/tex] be the Lie algebra consisting of 3x3 matrices of the form [tex]\begin{bmatrix} 0 & a & c \\ 0 & 0 & b \\ 0 & 0 & 0 \end{bmatrix}[/tex] where a,b,c are any complex numbers. The vector addition and scalar multiplication on [tex]\mathfrak{h}[/tex] are the usual operations on matrices. The Lie bracket on [tex]\mathfrak{h}[/tex] is defined as the matrix commutator: [tex][X,Y] = XY - YX[/tex] for any [tex]X,Y \in \mathfrak{h}.[/tex] Prove that the Lie algebras [tex]\mathfrak{g}[/tex] and [tex]\mathfrak{h}[/tex] are isomorphic.
The Attempt at a Solution
Firstly, is this the definition for (i):
[tex]\varphi[/tex] is a homomorphism if [tex]\varphi [x,y] = [\varphi (x) , \varphi (y) ][/tex] for all [tex]x,y\in\mathfrak{g}\,?[/tex]
What is the definition for (ii)?
For (iii) presumably I first have to show that a mapping [tex]\varphi : \mathfrak{g} \to \mathfrak{h}[/tex] is a homomorphism? If so how do I show [tex]\varphi [x,y] = [\varphi (x) , \varphi (y) ]\,?[/tex]
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