Many $\mathbb{Z}$-graded Lie algebras $\mathfrak{g}$ over $\mathbb{C}$ we would like to study are non-degenerate in the sense that
- $\dim_{\mathbb{C}} \mathfrak{g}_n < \infty \ \forall n \in \mathbb{Z}$
- $\mathfrak{g}_0$ is abelian
- For generic $\lambda \in \mathfrak{g}_0^*$, $\forall n \in \mathbb{N}$ the bilinear form $\mathfrak{g_n} \otimes \mathfrak{g}_{-n} \rightarrow \mathbb{C}$ given by $$a \otimes b \mapsto \lambda ([a,b])$$ is non-degenerate.
This includes simple Lie algebras, Loop algebras, affine Kac-Moody algebras, the Heisenberg algebra, the Witt algebra, and the Virasoro algebra.
For any of these examples, can we find some topological space $X$ such that $\pi_{*+1}(X) \cong \mathfrak{g}_{\geq 0}$ under the Whitehead bracket?
