Is it permissible for the Imam to sit on a chair when prostrating if he has undergone surgery on his left knee, knowing that his wife cannot prostrate for the same reason, and this is during congregational prayer with his family?

It is permissible for you to lead your family in prayer while sitting, whether they pray behind you standing or sitting due to an excuse. This is the view of the Hanafis and Shafi'is, who cite as evidence the Hadith of Aisha (may Allah be pleased with her) that the Prophet (peace and blessings be upon him) prayed sitting in the last part of his life, and his Companions prayed standing behind him.