Fr\'echet Quantum Supergroups

In this paper, we introduce Fr'echet quantum supergroups and their representations. By using the universal deformation formula of the abelian supergroups R^{m|n} we construct various classes of Fr'echet quantum supergroups that are deformation of classical ones. For such quantum supergroups, we find an analog of Kac-Takesaki operators that are superunitary and satisfy the pentagonal relation.