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We construct a new representation of the infinite rank Lie algebra$a_{\infty}$ with central charge $c=1$ on the Fock space $\mathit{F^{\otimes\frac{1}{2}}}$ of a single neutral fermion. We show that $\mathit{F^{\otimes\frac{1}{2}}}$ is a direct sum of irreducible integrable highest weight modulesfor $a_{\infty}$ with central charge $c=1$. We prove that as $a_{\infty}$modules $\mathit{F^{\otimes \frac{1}{2}}}$ is isomorphic to the Fock space$\mathit{F^{\otimes 1}}$ of the charged free fermions. As a corollary we obtainthe decompositions of certain irreducible highest weight modules for$d_{\infty}$ with central charge $c=\frac{1}{2}$ into irreducible highestweight modules for $d_{\infty}$ with central charge $c=1$.

Representations of $a_{\infty}$ and $d_{\infty}$ with central charge 1 on the single neutral fermion Fock space $\mathit{F^{\otimes \frac{1}{2}}}$