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Central extensions of the algebra of formal pseudo-differential symbols via Hochschild (co)homology and quadratic symplectic Lie algebras. (English) Zbl 1432.17018

Summary: We describe the space of central extensions of the associative algebra \(\Psi_n\) of formal pseudo-differential symbols in \(n \geq 1\) 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 Lie (co)homology group \(H_{Lie}^1(\Psi_n)\) of \(\Psi_n\) 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.

MSC:

17B56 Cohomology of Lie (super)algebras
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
17B66 Lie algebras of vector fields and related (super) algebras
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