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On quasifree representations of infinite dimensional symplectic group. (English) Zbl 1062.22037
The authors study an infinite dimensional generalization of metaplectic representations for the symplectic group. The group Sp$$(\infty)$$ is the set of invertible operators $$g=1+A$$ such that $$g$$ preserves a symplectic form on an infinite dimensional vector space and $$A$$ is of finite rank. First, the authors present a class of unitary representations of the Lie algebra sp$$(\infty)$$ on GNS spaces of quasifree states of the canonical commutation relations algebra (CCR algebra). This representation is obtained in the same way as the metaplectic representation of finite dimensional groups. The obtained representation is a natural generalization of the Shale representation, and it is referred to as quasifree representation. From this representation one obtains several representations of the double covering of Sp$$(\infty)$$. The authors study the Fock space and exponential vectors, the structure of the Fock representations of sp$$(\infty)$$, the structure of non-Fock representations of sp$$(\infty)$$, the quasi-equivalence of quasifree representations of sp$$(\infty)$$ and metaplectic representations of Sp$$(\infty, P)$$.

##### MSC:
 22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties 22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) 53D50 Geometric quantization
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