Central extensions of the algebra of formal pseudo-differential symbols via Hochschild (co)homology and quadratic symplectic Lie algebras

We describe the space of central extensions of the associative algebra ${\mathrm{\Psi }}_n$ of formal pseudo-differential symbols in $n\ge 1$ independent variables using Hochschild (co)homology groups: we prove that the first Hochschild (co)homology group $H{H}^1({\mathrm{\Psi }}_n)$ is 2n-dimensional and we use this fact to calculate the first Lie (co)homology group $H_{Lie}^1({\mathrm{\Psi }}_n)$ of ${\mathrm{\Psi }}_n$ equipped with the Lie bracket induced by its associative algebra structure. As an application, we use our calculations to provide examples of infinite-dimensional quadratic symplectic Lie algebras.