We investigate the stability of the super-KMS property under deformations. We show that a family of continuous deformations of the superderivation in the quantum algebra yields a continuous family of deformed super-KMS functionals. These functionals define a family of cohomologous, entire cocycles.
We define a planar para algebra, which arises naturally from combining planar
algebras with the idea of $\mathbb{Z}_{N}$ para symmetry in physics. A
subfactor planar para algebra is a Hilbert space representation of planar
tangles with parafermionic defects, that are invariant under para isotopy. For
each $\mathbb{Z}_{N}$, we construct a family of subfactor planar para algebras
which play the role of Temperley-Lieb-Jones planar algebras. The first example
in this family is the parafermion planar para algebra (PAPPA). Based on this
example, we introduce parafermion Pauli matrices, quaternion relations, and
braided relations for parafermion algebras which one can use in the study of
quantum information. An important ingredient in planar para algebra theory is
the string Fourier transform (SFT), that we use on the matrix algebra generated
by the Pauli matrices. Two different reflections play an important role in the
theory of planar para algebras. One is the adjoint operator; the other is the
modular conjugation in Tomita-Takesaki theory. We use the latter one to define
the double algebra and to introduce reflection positivity. We give a new and
geometric proof of reflection positivity, by relating the two reflections
through the string Fourier transform.Comment: 41 page
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