The de Rham Cohomology of Compact Lie Groups

Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak{g}$. We prove that the de Rham cohomology $H_{\text{dR}}^{•}\left(G\right)$ is isomorphic to the Lie algebra cohomology $H^{•}\left(\mathfrak{g},•\right)$, and that both are isomorphic to the space ${\left({\left({\text{Λ}}^{•}\mathfrak{g}\right)}^{}\right)}^{\mathfrak{g}}$ of $\mathfrak{g}$-invariant forms on the exterior algebra. In other words, $H^{•}\left(\mathfrak{g},•\right)•H_{\text{dR}}^{•}\left(G\right)•{\left({\left({\Lambda }^{•}\mathfrak{g}\right)}^{}\right)}^{\mathfrak{g}}.$ This result reduces the geometric problem of computing $H_{\text{dR}}^{•}\left(G\right)$ to the algebraic problem. We use the above isomorphism to calculate the de Rham cohomology of several classical groups $SO\left(3\right)$, $SO\left(4\right)$, $SO\left(5\right)$, $SO\left(6\right)$, and $SU\left(3\right)$. We provide detailed computations for $SO\left(3\right)$. For the higher-dimensional cases $SO\left(4\right)$, $SO\left(5\right)$, $SO\left(6\right)$, and $SU\left(3\right)$, we utilize MATLAB to compute the de Rham cohomology groups efficiently.