I'm trying to derive the SO(2,1) algebra from the SO(3) algebra.(adsbygoogle = window.adsbygoogle || []).push({});

The generators for SO(3) are given by:

[tex]M^{ij}_{ab}=i*(\delta_{a}^{i}\delta_{b}^{j}-\delta_{b}^{i}\delta_{a}^{j}) [/tex]

where "a" represents the row, and "b" represents the column, and "ij" represents the generator (12=spin in 3 direction, 23=spin in 1 direction, etc.).

To get SO(2,1), I thought you just raised the "a" term by using the metric tensor:

[tex]M^{(ij)a}_{b}=i*(g^{ai}\delta_{b}^{j}-\delta_{b}^{i}g^{aj}) [/tex]

and took the commutators of the M^{ij}with each other to derive the Lie Algebra.

But when I do this I don't seem to get the Lie algebra of SO(2,1). I have some notes that say the Lie Algebra of SO(2,1) is given by:

[D,H]=-iH

[K,D]=-iK

[H,K]=2iD

where D, H, and K are the generators. I get nothing like that when I take the commutators of M^{12}, M^{23}, and M^{31}.

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# SO(2,1) Lie Algebra Derivation

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