Relative Fourier transforms and expectations on coideal subalgebras

TL;DR

This paper establishes a bijection between right coideal *-subalgebras and left module quotient *-coalgebras in algebraic compact quantum groups, characterizing when positive expectations exist via a Fourier transform.

Contribution

It introduces a new correspondence between coideal *-subalgebras and quotient *-coalgebras, and characterizes positivity of expectations using a Fourier transform in quantum groups.

Findings

Bijection between coideal *-subalgebras and quotient *-coalgebras

Positivity of expectations linked to invariance under squared antipode

Fourier transform characterizes elements of quantum groups as functions on coalgebras

Abstract

For an algebraic compact quantum group $H$ we establish a bijection between the set of right coideal $*$-subalgebras $A\to H$ and that of left module quotient $*$-coalgebras $H\to C$. It turns out that the inclusion $A\to H$ always splits as a map of right $A$-modules and right $H$-comodules, and the resulting expectation $E:H\to A$ is positive (and lifts to a positive map on the full $C^*$ completion on $H$) if and only if $A$ is invariant under the squared antipode of $H$. The proof proceeds by Tannaka-reconstructing the coalgebra $C$ corresponding to $A\to H$ by means of a fiber functor from $H$-equivariant $A$-modules to Hilbert spaces, while the characterization of those $A\to H$ which admit positive expectations makes use of a Fourier transform turning elements of $H$ into functions on $C$.