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  1. #1
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    centre

    why is it we say that the centre of SU(N) is isomorphic to Z (mod n) and center of U(N) is isomorphic to U(1) when
    both their centres are {aI : lal=1}

    are they different or the same
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by alexandrabel90 View Post
    why is it we say that the centre of SU(N) is isomorphic to Z (mod n) and center of U(N) is isomorphic to U(1) when
    both their centres are {aI : lal=1}

    are they different or the same
    Well it's almost always true (as a rule of thumb--not a theorem) that given a (complex, although more general fields of characteristic zero work) matrix group G one has that \mathcal{Z}(G)=G\cap \left\{zI:z\in\mathbb{C}\right\} and this is true for \text{SU}(n) so that \mathcal{Z}\left(\text{SU}(n)\right)=\left\{zI:z\i  n\mathbb{C}\right\}\cap\text{SU}(n). That said, we know that since zI\in\text{SU}(n) it's true that 1=\det(zI)=z^n so that z_n\in\mu_n (this is a common notation for the n^{\text{th}} roots of unity) and moreover you can check that zI\in\mathcal{Z}\left(\text{SU}\left(n\right)\righ  t) for every z\in\mu_n so that \mathcal{Z}\left(\text{SU}(n)\right)\cong\mu_n but it's basic group theory that \mu_n\cong\mathbb{Z}_n. Does that help?
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