An Imprimitivity Theorem for finite algebraic quantum groups

Abstract

Let $\mathcal{G}$ be an algebraic quantum group and $\mathcal{U}$ a compact quantum subgroup. Given a left $\hat{\mathcal{U}}$-module algebra A with unit, we can endow $A\otimes \mathcal{G}$ with a structure of a right $\hat{\mathcal{U}}$-module algebra. The algebra of invariants for this action $(A\otimes \mathcal{G}{)}^{\hat{\mathcal{U}}}$ has a left action of $\hat{\mathcal{G}}$. We prove that for finite $\mathcal{G}$, $(A\otimes \mathcal{G}{)}^{\hat{\mathcal{U}}}\#\hat{\mathcal{G}}$ is Morita equivalent to $A\#\hat{\mathcal{U}}$.