Infinite dimensional metagonal and metaplictie groups I. The general notions and the metagonal group

She groups in question are central extensions by $S^1$ of groups of orthogonal and symplectic operators with Hilbert-Schmidt antilinear part acting in a Hilbert space. A thorough study of the corresponding 2-coeycles on their Lie algebras is presented. A limiting procedure relating the infinite dimensional groups with the corresponding finite dimensional ones is exhibited. Central extensions in the former case are shown to be nontrivial even on the topological level. Possible applications include unified models of the basic moduli for the Kac–Moody Lie algebras.