The short answer to this is that Alexander the Great's empire was split up after his death because no one person had the credibility or the ability to keep the whole thing together.
Specifically, Alexander's empire was split into four of what are called Hellenistic kingdoms. These were:
- Macedonia, which was ruled by the Antigonids.
- Syria and the eastern parts of the empire. These were ruled by the Seleucid Dynasty.
- Pergamum, in Asia Minor, ruled by the Attalids. This kingdom broke away from the Seleucids.
- Egypt, ruled by the Ptolemies.
All of these except Pergamum were founded by one or another of Alexander's generals or (in the case of Macdonia) by a descendant of a general.