I have a question in seeing this
$$U(n)=\frac{U(n)}{U(n-1)} \* \frac{U(n-1)}{U(n-2)}\*\cdots \*\frac{U(2)}{U(1)}\*U(1)$$


So, group U(n) is written as product of quotient spaces.


Is quotient space, for example $\frac{U(n)}{U(n-1)}$ , as topological space the same as
quotient gropup i.e. set of cosets?


How to prove $\frac{U(n)}{U(n-1)}$ is diffeomorphic to some sphere ?
Thx



Group or manifold ?
I'm not sure what you mean by the star product but yes, you see a quotient G/H as the set of cosets. There are topological requirements on H though, for instance being closed etc... at least for G/H to be a differentiable manifold, I'm not sure about topological manifold.


Finally, the idea to prove the sphere statement is to notice that U(n) acts transitively on a sphere of appropriate dimension, choose a point and prove that its stabilizer is U(n-1) somehow seen inside U(n) (e.g. lower block of a matrix in U(n))


You can find more details on both my answers in the Lie Groups chapter in Warner's book (Foundations of diff geom I think)


David


