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 Ψ_n of formal pseudo-differential symbols in n≥1 independent variables using Hochschild (co)homology groups: we prove that the first Hochschild (co)homology group HH¹(Ψ_n) is 2n-dimensional and we use this fact to calculate the first Lie (co)homology group H_Lie ¹(Ψ_n) of Ψ_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.