# Invariant Markov semigroups on quantum homogeneous spaces

## TL;DR

This paper studies invariant quantum Markov semigroups on quantum homogeneous spaces, classifies their generators as Laplace operators, and computes spectral dimensions for classical and quantum spheres.

## Contribution

It establishes one-to-one correspondences between invariant functionals and quantum Markov semigroups, providing a classification framework for these semigroups on quantum spaces.

## Key findings

- Classified generators of Markov semigroups on spheres
- Computed spectral dimensions for classical and quantum spheres
- Established correspondences between invariant functionals and semigroups

## Abstract

Invariance properties of linear functionals and linear maps on algebras of functions on quantum homogeneous spaces are studied, in particular for the special case of expected coideal *-subalgebras. Several one-to-one correspondences between such invariant functionals are established. Adding a positivity condition, this yields one-to-one correspondences of invariant quantum Markov semigroups acting on expected coideal *-subalgebras and certain convolution semigroups of states on the underlying compact quantum group. This gives an approach to classifying invariant quantum Markov semigroups on these quantum homogeneous spaces. The generators of these semigroups are viewed as Laplace operators on these spaces. The classical sphere $S^{N-1}$, the free sphere $S^{N-1}_+$, and the half-liberated sphere $S^{N-1}_*$ are considered as examples and the generators of Markov semigroups on these spheres a classified. We compute spectral dimensions for the three families of spheres based on the asymptotic behaviour of the eigenvalues of their Laplace operator.