Planar Para Algebras, Reflection Positivity
Abstract
We define a planar para algebra, which arises naturally from combining planar algebras with the idea of 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 Z_{N}, we construct a family of subfactor planar para algebras that play the role of TemperleyLiebJones 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), which 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 TomitaTakesaki 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.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 May 2017
 DOI:
 10.1007/s0022001627794
 arXiv:
 arXiv:1602.02662
 Bibcode:
 2017CMaPh.352...95J
 Keywords:

 Mathematics  Quantum Algebra;
 Condensed Matter  Mesoscale and Nanoscale Physics;
 High Energy Physics  Theory;
 Mathematical Physics;
 Mathematics  Operator Algebras
 EPrint:
 41 pages