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Prove that there is an isomorphism between Lie superalgebras sl(2|1) and sl(1|2) ?

Prove that there is an isomorphism between Lie superalgebras sl(2|1) and sl(1|2) ?
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Answer #1

Let us take a function defined as the following:

where

Note that A will be in sI(2|1) if and only if B is in sI(1|2) (this happens because both have supertrace 0, you can see this by looking at the block matrix form). Now from the definition of the function, we can see that it is surjective. And also the inverse exists as well. Thus it is a bijection. Also note that it is closed under matrix addition and scalar multiplication. Thus it is an isomorphism.

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