The generators ##(A_{ab})_{st}## of the ##so(n)## Lie algebra are given by:(adsbygoogle = window.adsbygoogle || []).push({});

##(A_{ab})_{st} = -i(\delta_{as}\delta_{bt}-\delta_{at}\delta_{bs}) = -i\delta_{s[a}\delta_{b]t}##,

where ##a,b## label the number of the generator, and ##s,t## label the matrix element.

Now, I need to prove the following commutation relation using the definition above:

##([A_{ij},A_{mn}])_{st} = -i(A_{j[m}\delta_{n]i}-A_{i[m}\delta_{n]j})_{st}##.

Here's my attempt.

##([A_{ij},A_{mn}])_{st}##

## = (A_{ij})_{sp}(A_{mn})_{pt}-(ij \iff mn)##

##= -\delta_{s[i}\delta_{j]p}\delta_{p[m}\delta_{n]t}-(ij \iff mn)##

##= -\delta_{s[i}\delta_{j][m}\delta_{n]t}+\delta_{s[m}\delta_{n][i}\delta_{j]t}##

Could you please suggest the next couple of steps? Should I expand all the antisymmetrised Kronecker delta's, or is there some sneaky shortcut to get to the answer?

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# Deriving the commutation relations of the so(n) Lie algebra

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