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?

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

Loading...

Similar Threads - Deriving commutation relations | Date |
---|---|

I Splitting ring of polynomials - why is this result unfindable? | Feb 11, 2018 |

A Second derivative of a complex matrix | Jan 6, 2018 |

I How to find the matrix of the derivative endomorphism? | Oct 22, 2017 |

I Entries in a direction cosine matrix as derivatives | Feb 20, 2017 |

B Functional derivative | Oct 28, 2016 |

**Physics Forums - The Fusion of Science and Community**