Triviality of vector bundles over Lie groups

Question

I was wondering whether the triviality of the tangent bundle of a Lie group is a shared feature with all other vector bundles over Lie groups.

Naturally, the first example of a non-trivial vector bundle that comes to mind is the Moebius strip, which is a vector bundle over $S^1$ -a Lie group-, so the answer is no. However, the tangent bundle of a Lie group is also a group, and is hence orientable (unlike the Moebius strip).

So, my question is: Are orientable vector bundles over Lie groups trivial? or actually better, does anybody know an example of a non-trivial vector bundle over a Lie group which is itself a Lie group?

I thought that a way around this question was to inspect the Euler class of the given bundle, because I thought that I knew the cohomology of Lie groups. I had Borel's theorem in mind:

"If $G$ is a compact connected Lie group, then the cohomology ring is an exterior algebra of odd degree"

I thought that this implied that even rank bundles would be necessarily trivial. Unfortunately, later I realized this doesn't say that there are no elements of even degree... Also, this would only work for compact groups, and I would like to say something of non-compact groups as well.

Answer 1

Here's an indirect way to see that most compact connected Lie groups have nontrivial complex vector bundles over them. The starting point is the observation that if $X$ is a finite CW complex then taking Chern characters gives an isomorphism

$$K(X) \otimes \mathbb{Q} \cong H^{2 \bullet}(X, \mathbb{Q})$$

between the rationalized complex K-theory of $X$ and the rational even cohomology of $X$. Now, almost all compact connected Lie groups have nontrivial rational even cohomology: perhaps the simplest example is $S^1 \times S^1$, and the simplest simply connected example is $SU(2) \times SU(2) \cong \text{Spin}(4)$, which has rational cohomology $\mathbb{Q}[x_3, y_3]$ where $x_3, y_3$ are odd, and hence their product $x_3 y_3$ is even. Because the Chern character isomorphism is defined in terms of Chern classes, the conclusion is that not only does there exist a nontrivial complex vector bundle on $SU(2) \times SU(2)$, but there exists one with nontrivial Chern class $c_3$.

More explicitly, the classification of complex line bundles over $S^1 \times S^1$ is given by $H^2(S^1 \times S^1, \mathbb{Z}) \cong \mathbb{Z}$, so there are countably many nontrivial complex line bundles over $S^1 \times S^1$. They can all be described as holomorphic line bundles over elliptic curves in terms of divisors and meromorphic functions, if you like.

I know less about real vector bundles; I expect most compact connected Lie groups have nontrivial real K-theory (even the simply connected ones, in which case every vector bundle is orientable), but I don't know how to prove that off the top of my head.