Lie algebra cohomlogy of $\mathbb{R}^2$

Question

Consider $\mathbb{R}^2$ as a real Lie algebra. How can we prove that $\mathcal{H}^2 (\mathbb{R}^2)\cong \mathbb{R}$, the second cohomology space of $\mathbb{R}^2$?

I really appreciate if someone could help me about it. Thanks in advance.

Answer 1

For an abelian Lie algebra $L$ over a field $K$, the second cohomology space $H^2(L,K)$, by definition of the Chevalley-Eilenberg complex, is just given by ${\rm Hom} (\Lambda^2 (L),K)$ which has dimension $\binom{d}{2}$, where $d=\dim(L)$.