TITLE: Cohomology of the unitary group
QUESTION [8 upvotes]: 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?

REPLY [10 votes]: 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.