  The Ashtekar and Ashtekar-Barbero connection variable formulations of Kerr
isolated horizons are derived. Using a regular Kinnersley tetrad in
horizon-penetrating Kruskal-Szekeres-like coordinates, the spin coefficients of
Kerr geometry are determined by solving the first Maurer-Cartan equation of
structure. Isolated horizon conditions are imposed on the tetrad and the spin
coefficients. A transformation into an orthonormal tetrad frame that is fixed
in the time gauge is applied and explicit calculations of the spin connection,
the Ashtekar and Ashtekar-Barbero connections, and the corresponding curvatures
on the horizon 2-spheres are performed. Since the resulting Ashtekar-Barbero
curvature does not comply with the simple form of the horizon boundary
condition of Schwarzschild isolated horizons, i.e., on the horizon 2-spheres,
the Ashtekar-Barbero curvature is not proportional to the Plebanski 2-form,
which is required for an SU(2) Chern-Simons treatment of the gauge degrees of
freedom in the horizon boundary in the context of loop quantum gravity, a
general method to construct a new connection whose curvature satisfies such a
relation for Kerr isolated horizons is introduced. For the purpose of
illustration, this method is employed in the framework of slowly rotating Kerr
isolated horizons.
