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Many $\mathbb{Z}$-graded Lie algebras $\mathfrak{g}$ over $\mathbb{C}$ we would like to study are non-degenerate in the sense that

  1. $\dim_{\mathbb{C}} \mathfrak{g}_n < \infty \ \forall n \in \mathbb{Z}$
  2. $\mathfrak{g}_0$ is abelian
  3. 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?

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You probably know that the answer is yes if you replace $\mathfrak{g}_{\geq 0}$ with $\mathfrak{g}_{\geq 1}$. This is actually true for any positively graded Lie algebra. –  Fernando Muro Feb 16 '12 at 18:39

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