As is well known, one can view $\mathbb{CP}^n$ as a quotient of the unit $(2n + 1)$-sphere in $\mathbb{C}^{n+1}$ under the action of $U(1)$, since every line in $\mathbb{C}^{n+1}$ intersects the unit sphere in a circle. 


Moreover, we have $S^{2n + 1} = SU(n+1)/SU(n)$, where $SU(n)$ embeds into the bottom right-hand corner (say).


My question is: Is there an embedding $j$ of $U(1)$ into $SU(n+1)/SU(n)$ that gives the $U(1)$ action as a left (right) multiplication, i.e. such that $A.e^{i \theta} = Aj(e^{i \theta})$, for all $A \in SU(n+1)/SU(n)$?


For $n=1$, it's easy: $SU(2) = S^3$, and we embed $e^{i\theta}$ as  

$\left(
 \begin{array}{cc}
 e^{i \theta} & 0 \\\\
 0 & e^{-i \theta}
 \end{array}
\right)$.



Complex Projective Space as a $U(1)$ quotient
The U(1) group is the torus in SU(N+1) which commutes with SU(N). In the example
given in the question where SU(N) is chosen as the bottom N-dimensional block. U(1) consists
of the diagonal matrices 


diag{exp(N\*i\*theta), exp(-i\*theta), . . . . (N-times) exp(-i\*theta)}


Please observe that the restriction to the bottom N-dimensional block is proportional to the unit matrix thus it commutes with the whole of SU(N), also it belongs to SU(N+1), since its has a unit determinant.


The reason that the U(1) and the SU(N) factors commute is due to a theorem by [A. Borel](http://www.pnas.org/content/40/12/1147.full.pdf+html) which states that the denominator subgroup of homogeneous Kaehlerian spaces must be the centralizer of a torus. In our case the torus is the U(1) subgroup and the certralizer is SU(N)\*U(1)


