Symmetry operators and the separability of massive Klein-Gordon and Dirac equations in the general five-dimensional Kerr-(anti-)de Sitter black hole background

It is shown that the Dirac equation is separable by variables in a five-dimensional rotating Kerr-(anti-)de Sitter black hole with two independent angular momenta. A first-order symmetry operator that commutes with the Dirac operator is constructed in terms of a rank-3 Killing-Yano tensor whose square is a second-order symmetric Staeckel-Killing tensor admitted by the five-dimensional Kerr-(anti-)de Sitter spacetime. We highlight the construction procedure of such a symmetry operator. In addition, the first law of black hole thermodynamics has been extended to the case that the cosmological constant can be viewed as a thermodynamical variable.