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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- May 13th 2011, 09:39 AMalexandrabel90centre
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 - May 13th 2011, 07:10 PMDrexel28
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 $\displaystyle G$ one has that $\displaystyle \mathcal{Z}(G)=G\cap \left\{zI:z\in\mathbb{C}\right\}$ and this is true for $\displaystyle \text{SU}(n)$ so that $\displaystyle \mathcal{Z}\left(\text{SU}(n)\right)=\left\{zI:z\i n\mathbb{C}\right\}\cap\text{SU}(n)$. That said, we know that since $\displaystyle zI\in\text{SU}(n)$ it's true that $\displaystyle 1=\det(zI)=z^n$ so that $\displaystyle z_n\in\mu_n$ (this is a common notation for the $\displaystyle n^{\text{th}}$ roots of unity) and moreover you can check that $\displaystyle zI\in\mathcal{Z}\left(\text{SU}\left(n\right)\righ t)$ for every $\displaystyle z\in\mu_n$ so that $\displaystyle \mathcal{Z}\left(\text{SU}(n)\right)\cong\mu_n$ but it's basic group theory that $\displaystyle \mu_n\cong\mathbb{Z}_n$. Does that help?