  We introduce a notion of Ricci curvature lower bound for symmetric
sub-Markovian semigroups. We use this notion to investigate functional calculus
of the Hodge-Dirac operator associated to the semigroup in link with the
boundedness of suitable Riesz transforms. Our paper offers a unified framework
that not only encapsulates existing results in some contexts but also yields
new findings in others. This is demonstrated through applications in the
frameworks of Riemannian manifolds, compact (quantum) groups, noncommutative
tori, Ornstein-Uhlenbeck semigroup, $q$-Ornstein-Uhlenbeck semigroups and
semigroups of Schur multipliers. We also provide an $\mathrm{L}^p$-Poincar\'e
inequality that is applicable to all previously discussed contexts under
assumptions of boundedness of Riesz transforms and uniform exponential
stability. Finally, we prove the boundedness of some Riesz transforms in some
contexts as compact Lie groups.
