The planar algebra of a coaction
Résumé
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.