su(n) is isomorphic to sl(n,C), when we tensor su(n) with the complex numbers we get sl(n,C).(adsbygoogle = window.adsbygoogle || []).push({});

Say we have su(2) with E_1= 1/2 [i, 0;0, -i], E_2=1/2[0,1;-1,0], E_3=1/2[0, i; i,0]

sl(2,C) with F_1=[1, 0; 0, -1], F_2=[0, 1; 0, 0], F_3=[0, 0; 1, 0]

so that [E_1, E_2]=E_3, [E_2, E_3]= E_1, [E_3, E_1]=E_2

and [F_1, F_2]=2F_3, [F_1, F_3]=-2F_3, [F_2,F_3]=F_1

Now I could write F_1=-2E_1, F_2=E_2-iE_3, F_3=E_2+iE_3, which i guess means tensoring su(2) with complex numbers and by what I get from the su(2) bracket relations to the sl(2,C) bracket relations.

But where is the isomorphism?

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# Su(n) to sl(n,C)

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