- #1
spaghetti3451
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The commutation relations for the ##\mathfrak{so(n)}## Lie algebra is:##([A_{ij},A_{mn}])_{st} = -i(A_{j[m}\delta_{n]i}-A_{i[m}\delta_{n]j})_{st}##.where the generators ##(A_{ab})_{st}## of the ##\mathfrak{so(n)}## Lie algebra are given by:##(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.I would like to show that the structure constants ##f_{ij,mn}^{ks}## of the ##\mathfrak{so(n)}## Lie algebra such that##[A_{ij},A_{mn}] = if_{ij,mn}^{ks}A_{ks}##are given by##f_{ij,mn}^{ks} = \delta_{k[j}\delta_{i][m}\delta_{n]s}##.Can someone help me out with this?
where ##a,b## label the number of the generator, and ##s,t## label the matrix element.I would like to show that the structure constants ##f_{ij,mn}^{ks}## of the ##\mathfrak{so(n)}## Lie algebra such that##[A_{ij},A_{mn}] = if_{ij,mn}^{ks}A_{ks}##are given by##f_{ij,mn}^{ks} = \delta_{k[j}\delta_{i][m}\delta_{n]s}##.Can someone help me out with this?