  This paper studies the continuous-time q-learning in the mean-field
jump-diffusion models from the representative agent's perspective. To overcome
the challenge when the population distribution may not be directly observable,
we introduce the integrated q-function in decoupled form (decoupled
Iq-function) and establish its martingale characterization together with the
value function, which provides a unified policy evaluation rule for both
mean-field game (MFG) and mean-field control (MFC) problems. Moreover,
depending on the task to solve the MFG or MFC problem, we can employ the
decoupled Iq-function by different means to learn the mean-field equilibrium
policy or the mean-field optimal policy respectively. As a result, we devise a
unified q-learning algorithm for both MFG and MFC problems by utilizing all
test policies stemming from the mean-field interactions. For several examples
in the jump-diffusion setting, within and beyond the LQ framework, we can
obtain the exact parameterization of the decoupled Iq-functions and the value
functions, and illustrate our algorithm from the representative agent's
perspective with satisfactory performance.
