The Kashiwara involution in the general Kac–Moody–Borcherds case

Kashiwara (1993) [10], proved the existence of an involution – called the Kashiwara involution – in the crystal B(∞) of , where g is a symmetrizable Kac–Moody algebra. Recently Joseph (in press) [3], gave a purely combinatorial proof in the not necessarily symmetrizable Kac–Moody case. In this paper we prove the existence of a Kashiwara involution in the not necessarily symmetrizable Kac–Moody–Borcherds case following Joseph (in press) [3] and we prove an additivity property for B(∞) in the purely imaginary case.