  It is well-known that irreversible MCMC algorithms converge faster to their
stationary distributions than reversible ones. Using the special geometric
structure of Lie groups $\mathcal G$ and dissipation fields compatible with the
symplectic structure, we construct an irreversible HMC-like MCMC algorithm on
$\mathcal G$, where we first update the momentum by solving an OU process on
the corresponding Lie algebra $\mathfrak g$, and then approximate the
Hamiltonian system on $\mathcal G \times \mathfrak g$ with a reversible
symplectic integrator followed by a Metropolis-Hastings correction step. In
particular, when the OU process is simulated over sufficiently long times, we
recover HMC as a special case. We illustrate this algorithm numerically using
the example $\mathcal G = SO(3)$.
