  We examine the quantum phase diagram of the fractional quantum Hall effect in
the lowest Landau level in half-filled bilayer structures as a function of
tunneling strength and layer separation. Using numerical exact diagonalization
we investigate the important question of whether this system supports a
fractional quantum Hall effect described by the non-Abelian Moore-Read Pfaffian
state in the strong tunneling regime. We find that, although it is in principle
possible, it is unlikely that the non-Abelian FQHE exists in the lowest Landau
level. We establish that all so far observed FQHE states in half-filled lowest
Landau level bilayers are most likely described by the Abelian Halperin 331
state.
