In many quantum field theory textbooks, when looking for the Lie algebra of the Poincaré group, $$[M^{\mu\nu}, M^{\rho\tau}]=i(g^{\mu\tau}M^{\nu\rho}+g^{\nu\rho}M^{\mu\tau}-g^{\mu\rho}M^{\nu\tau}-g^{\nu\tau}M^{\mu\rho})$$ $$[P^{\mu}, M^{\nu\rho}]=-i(g^{\mu\rho}P^{\nu}-g^{\mu\nu}P^{\tau})$$ $$[P^{\mu},P^{\nu}]=0$$ they use a particular unitary representation to check that these commutation relations hold under the unitary representation, and thereby declare that these algebraic relations hold in the general case.
Here I have two questions.
- when checking the commutatiton relation of a Lie algebra, is it enough to take a particular representation and check the exchange relation for it?
I thought that when constructing a representation of a Lie algebra, the commutation relations between generators must be known in advance to construct the representation, but am I mistaken?
- Is it possible to calculate the commutation relations between generators of a Lie algebra in such a way that does not depend on a particular representation is used?