Crystal bases of modified $\imath$quantum groups of certain quasi-split types

Abstract: In order to see the behavior of $\imath$canonical bases at $q = \infty$, we introduce the notion of $\imath$crystals associated to an $\imath$quantum group of certain quasi-split type. The theory of $\imath$crystals clarifies why $\imath$canonical basis elements are not always preserved under natural homomorphisms. Also, we construct a projective system of $\imath$crystals whose projective limit can be thought of as the $\imath$canonical basis of the modified $\imath$quantum group at $q = \infty$.