Planar functions and related group algebras

Some Planar Algebras Related to Graphs

Let X denote a finite nonempty set, and let W denote a matrix whose rows and columns are indexed by X and whose entries belong to some field K. We study three planar algebras related to W . Briefly, a planar algebra is a graded vector space V = ∪n∈Z+∪{+,−}Vn which is closed under “planar” operators. The first planar algebra which we study, FW = ∪FW n , is defined by the group theoretic properti...

Permutation Group Algebras and Parking Functions

Group invariants in algebras of generalized functions

We study invariance properties of Colombeau generalized functions under actions of smooth Lie transformation groups. Several characterization results analogous to the smooth setting are derived and applications to generalized rotational invariance are given. Mathematics Subject Classification (2000): Primary: 46F30; secondary: 58E40,46T30, 35A30

Subfactors and Planar Algebras

An inclusion of II1 factors N ⊂ M with finite Jones index gives rise to a powerful set of invariants that can be approached successfully in a number of different ways. We describe Jones’ pictorial description of the standard invariant of a subfactor as a so-called planar algebra and show how this point of view leads to new structure results for subfactors. 2000 Mathematics Subject Classificatio...

Free Analysis and Planar Algebras

We study 2-cabled analogs of Voiculescu’s trace and free Gibbs states on Jones planar algebras. These states are traces on a tower of graded algebras associated to a Jones planar algebra. Among our results is that, with a suitable de nition, niteness of free Fisher information for planar algebra traces implies that the associated tower of von Neumann algebras consists of factors, and that the s...