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On planar algebras arising from hypergroups

R. M. Green
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Abstract

Let $A$ be an associative algebra with identity and with trace. We study the family of planar algebras on 1-boxes that arise from $A$ in the work of Jones, but with the added assumption that the labels on the 1-boxes come from a discrete hypergroup in the sense of Sunder. This construction equips the algebra $P_n^A$ with a canonical basis, $\BB_n^A$, which turns out to be a ``tabular'' basis. We examine special cases of this construction to exhibit a close connection between such bases and Kazhdan--Lusztig bases of Hecke algebras of types $A$, $B$, $H$ or $I$.

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R. M. Green. 2002-09-18. On planar algebras arising from hypergroups. https://arxiv.org/abs/math/0209224

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