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

摘要

We describe the space of central extensions of the associative algebra psi(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 H H-1(psi(n)) is 2n-dimensional and we use this fact to calculate the first Lis (co)homology group H-Lie(1)(psi(n)) of psi(n) equipped with the Lie bracket induced by it's associative algebra structure. As an application, we use our calculations to provide examples of infinite-dimensional quadratic symplectic Lie algebras. (C) 2017 Elsevier B.V. All rights reserved.