  In this article we will analyze the possibility of a nontrivial central
extension of the Poisson algebra of the diffeomorphism generators, which
respect certain boundary conditions on the black hole bifurcation. The origin
of a possible central extension in the algebra is due to the existence of
boundary terms in the in the canonical generators. The existence of such
boundary terms depend on the exact boundary conditions one takes. We will check
two possible boundary conditions i.e. fixed bolt metric and fixed surface
gravity. In the case of fixed metric the the action acquires a boundary term
associated to the bifurcation but this is canceled in the Legendre
transformation and so absent in the canonical generator and so in this case the
possibility of a nontrivial central extension is ruled out. In the case of
fixed surface gravity the boundary term in the action is absent but present in
the Hamiltonian. Also in this case we will see that there is no nontrivial
central extension, also if there exist a boundary term in the generator.
