Abstract: The present invention provides a catalyst for polymerizing an .alpha.-olefin which is capable of producing an .alpha.-olefin polymer having a high molecular weight in a high yield under the polymerization conditions including at high temperature, and a process for preparing an .alpha.-olefin polymer, which catalyst comprises a Component (A) which comprises a coordination compound having a skeleton of a metallocene compound comprising an indenyl ring or a hydroindenyl ring which is an indenyl having a six-membered portion hydrogenated as a ligand and having Ti, Zr or Hf as a metal component in combination with a Component (B) which is, for example, an alumoxane. The coordination compound is crosslinked at the six-membered rings characterized by the fact that two of the indenyl or hydroindenyl groups are crosslinked at the six-membered ring of the indenyl or hydroindenyl group.