The de-Rham cohomology ring of U(n) is the exterior algebra generated by the odd-dimensional classes x\_1, x\_3, ..., x\_(2n-1). Moreover, on a Lie group every cohomology class is represented by a unique invariant form (both left and right). I ask two questions:
1) if we represent U(n) with matrices U = [z\_ij], what is an explicit expression of a generator as an invariant form of U(n), in terms of the differentials dz\_ij, for each odd degree between 1 and 2n-1?
2) in the 1-dimensional space of the invariant forms of a fixed degree (multiples of x\_i), which are the two (opposite to each other) which represent the real image of a generator of the integral cohomology?



Cohomology of the unitary group
Here's an answer to the first question.


Let $\theta = U^{-1} dU$ be the left-invariant Maurer-Cartan one-form: it is a matrix of one-forms. Then
$$\omega\_{2n+1} = \mathrm{Tr} \theta^{2n+1}$$
is the desired bi-invariant form. Here $\theta^{2n+1}$ stands both for the wedge and matrix product of $\theta$ with itself $2n+1$ times.


I'm not sure I understand the second question: what "two" representatives are you talking about? I think the canonical representative will be certain multiple of $\omega\_{2n+1}$. I'll try to fish out a reference later on.


