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 Incoming links: Lorentz algebra in three spacetime dimensions

Differentiate the defining relation at

Λ = I + tX + O (t^{2})

. The linear term gives the Lie algebra

g = {X \in M_{3} (R) : X^{T} η + η X = 0} .

Solving this equation between matrices yields

X = 0 a b a 0 c b - c 0 = a K_{1} + b K_{2} + c J,

where a convenient basis is

K_{1} = 010100000, K_{2} = 001000100, J = 000001 0 - 1 0 .

The first two generate Lorentz boosts and the last a spatial rotation. Direct commutators give

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

There are three independent real coefficients, so

dim_{R} g = 3

R J

, and the rank of a semisimple Lie algebra is one. Explicitly, if

X = a K_{1} + b K_{2} + c J

and

Y = a^{'} K_{1} + b^{'} K_{2} + c^{'} J

, their commutator vanishes exactly when

b c^{'} - c b^{'} = c a^{'} - a c^{'} = b a^{'} - a b^{'} = 0.

For nonzero

X

Y

to be proportional to

X

, so there is no two-dimensional commuting subalgebra. The split real rank is also one, realized by the semisimple Lorentz boost generator

K_{1}

. Thus

g = so (1, 2), dim g = 3, rank g = 1.

For this Lorentz algebra in three spacetime dimensions, disconnected components of the full Lorentz group do not change its tangent Lie algebra at the identity.