Splitting object in half

I’m using an existing 3D-model that I imported to Blender. This model contains a cone and I now need to split this cone in to two separate objects right in the middle. This splitting shouldn’t be visible in the 3D-model, I merely need to name the two halfs of the cone differently. How can I split the cone?

Select half of your cone and press “P”, choose “selected”.

Ok, Now I got the mesh separated. How do I create a new object from these?

do you mean there should be a modelled split, or just a split for selection?
if modelled, take your seperated halfs and join them (ctrl+J)
if selectable, do the same and then remove doubles (W>remove doubles), then select on half, go to the mesh buttons, and press (Vertex Groups>New>assign) then do the same for the other side. remember to name the groups.