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Luck0
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Let U ∈ SU(N) and {ta} be the set of generators of su(N), a = 1, ..., N2 - 1. The action of the adjoint representation of U on some generator ta can be written as
Ad(U)ta = Λ(U)abtb
I want to characterize the matrix Λ(U), i. e., I want to see which of its elements are independent. It's known that it belongs to a subspace of SO(N2 - 1), and we can derive some constraints by saying that the adjoint action respects the Lie bracket. Here https://mathoverflow.net/questions/179032/characterising-the-adjoint-representation-of-sun it says that the most general characteristic of the adjoint representation is that it preserves some 3-form, but I can't find the details (I don't really know what am I looking for). Does you guys know some place where I can find details about this kind of characterization of the adjoint representation for compact groups? Thanks!
Ad(U)ta = Λ(U)abtb
I want to characterize the matrix Λ(U), i. e., I want to see which of its elements are independent. It's known that it belongs to a subspace of SO(N2 - 1), and we can derive some constraints by saying that the adjoint action respects the Lie bracket. Here https://mathoverflow.net/questions/179032/characterising-the-adjoint-representation-of-sun it says that the most general characteristic of the adjoint representation is that it preserves some 3-form, but I can't find the details (I don't really know what am I looking for). Does you guys know some place where I can find details about this kind of characterization of the adjoint representation for compact groups? Thanks!