A duality between non-compact semisimple symmetric pairs and commutative compact semisimple symmetric triads and its general theory

The present paper investigates a natural generalization of the duality between Riemannian symmetric pairs of compact type and those of non-compact type `a la 'E. Cartan. The main result of this paper is to construct an explicit description of a one-to-one correspondence between non-compact pseudo-Riemannian semisimple symmetric pairs and commutative compact semisimple symmetric triads, which is called the duality theorem. Further, we develop a general theory of the duality theorem.