The planar algebra of a coaction

Abstract

We study actions of ``compact quantum groups'' on ``finite quantum spaces''. According to Woronowicz and to general $\c^*$-algebra philosophy these correspond to certain coactions $v:A\to A\otimes H$. Here $A$ is a finite dimensional $\c^*$-algebra, and $H$ is a certain special type of Hopf *-algebra. If $v$ preserves a positive linear form $\phi :A\to\c$, a version of Jones' ``basic construction'' applies. This produces a certain $\c^*$-algebra structure on $A^{\otimes n}$, plus a coaction $v_n :A^{\otimes n}\to A^{\otimes n}\otimes H$, for every $n$. The elements $x$ satisfying $v_n(x)=x\otimes 1$ are called fixed points of $v_n$. They form a $\c^*$-algebra $Q_n(v)$. We prove that under suitable assumptions on $v$ the graded union of the algebras $Q_n(v)$ is a spherical $\c^*$-planar algebra.