- #1
ElDavidas
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Homework Statement
Take
[tex] L = \left(\begin{array}{ccc}0 & -a & -b \\b & c & 0 \\a & 0 & -c\end{array}\right) [/tex]
where a,b,c are complex numbers.
Homework Equations
I find that a basis for the above Lie Algebra is
[tex]e_1 = \left(\begin{array}{ccc}0 & -1 & 0 \\0 & 0 & 0 \\1 & 0 & 0 \end{array}\right) [/tex]
[tex]e_2 = \left(\begin{array}{ccc}0 & 0 & -1 \\1 & 0 & 0 \\0 & 0 & 0 \end{array}\right) [/tex]
[tex]e_3 = \left(\begin{array}{ccc}0 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & -1 \end{array}\right) [/tex]
I then calculate all the products [itex] [e_i,e_j] [/itex] and see that L is non-abelian and simple
The Attempt at a Solution
The question then asks show L is isomorphic to sl(2,C). I have found [itex] e,f,h \in L [/itex] such that [itex] [h,e] = 2e, [h,f] = -2f, [e,f] = h[/itex]
where,
[tex]h = \left(\begin{array}{ccc}0 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & -2 \end{array}\right) [/tex]
[tex]e = \left(\begin{array}{ccc}0 & 0 & -\sqrt{2} \\ \sqrt{2} & 0 & 0 \\0 & 0 & 0 \end{array}\right) [/tex]
[tex]f = \left(\begin{array}{ccc}0 & -\sqrt{2} & 0 \\0 & 0 & 0 \\ \sqrt{2} & 0 & 0 \end{array}\right) [/tex]
if I haven't made any mistakes. I see L is homomorphic to sl(2,C) but how do I show it's an isomorphism? (i.e show the injection and surjection). I know the definitions for an injection map and a surjection map but don't know how to apply it in this case.
Thanks in advance for any help.
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