 Lorentz algebra in three spacetime dimensions

ID: lorentz-algebra-in-three-spacetime-dimensions

With signature

(-, +, +)

, two Lorentz boost generators and one spatial rotation obey

[J, K_{1}] = K_{2}

[J, K_{2}] = - K_{1}

and

[K_{1}, K_{2}] = - J

. Replacing the boosts by

i K_{1}, i K_{2}

gives the compact cyclic brackets, proving the complexified isomorphism with the SU(2) Lie algebra. The real algebra has dimension three and rank one. Its finite-dimensional complex irreducible representations have dimensions

m + 1

m

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