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At least for low values of $N$ like $2$ or $3$ and such I would like to know if there are explicit matrices known giving the representation of $u(N)$ or $U(N)$ in the adjoint?

(..a related query: Is it for the Lie group or the Lie algebra of U(N) that it is true that the weight vectors in the fundamental/vector representation can be taken to be N N-vectors such that all have weight/eigenvalue 1 under its Cartan and the ith of them has 1 in the ith place and 0 elsewhere and for the conjugate of the above representation its the same but now with (-1)?..I guess its for the u(N) since they are skew-Hermitian but would still like to know of a precise answer/proof..)

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    Just write down your favourite basis of the Lie algebra, and express conjugation by (your favourite) elements of $U(n)$ in that basis.2013-12-01
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    The most popular ones are the [generalized Gell-Mann and the Sylvester clock-and-shift matrices](https://en.wikipedia.org/wiki/Generalizations_of_Pauli_matrices).2017-12-21

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