Is it permissible for us to supplicate after the prayer, and is it permissible for us to supplicate together with the Imam after finishing the prayer?

Supplicating with raised hands after the obligatory prayers is an innovation (bid'ah), because it has not been transmitted from the Prophet (peace be upon him) nor from his Companions. However, supplicating without raising the hands is permissible. Raising the hands in supplication after concluding the obligatory prayer is contrary to the Sunnah (definition). Reciting supplications aloud in a collective manner, as a continuous practice, after the five daily prayers and the regular Sunnah prayers, is a reprehensible innovation.