Метаплектик топ және оның қолданыстары

In mathematics, the metaplectic group Mp2n is a double cover of the symplectic group Sp2n. It can be defined over either real or p adic numbers. The construction covers more generally the case of an arbitrary local or finite field, and even the ring of adeles. The metaplectic group has a particularly significant infinite dimensional linear representation, the Weil representation. It was used by André Weil to give a representation theoretic interpretation of theta functions, and is important in the theory of modular forms of half integral weight and the theta correspondence.

The fundamental group of the symplectic Lie group Sp2n(R) is infinite cyclic, so it has a unique connected double cover, which is denoted Mp2n(R) and called the metaplectic group. The metaplectic group Mp2(R) is not a matrix group: it has no faithful finite dimensional representations. Therefore, the question of its explicit realization is nontrivial. It has faithful irreducible infinite dimensional representations, such as the Weil representation described below. It can be proved that if F is any local field other than C, then the symplectic group Sp2n(F) admits a unique perfect central extension with the kernel Z/2Z, the cyclic group of order 2, which is called the metaplectic group over F.
It serves as an algebraic replacement of the topological notion of a 2 fold cover used when 1=F = R. The approach through the notion of central extension is useful even in the case of real metaplectic group, because it allows a description of the group operation via a certain cocycle.