Global crystal bases for integrable modules over a quantum symmetric pair of type AIII

Abstract

In this paper, we study basic properties of global j-crystal bases for integrable modules over a quantum symmetric pair coideal subalgebra U j associated to the Satake diagram of type AIII without black nodes. Also, we obtain an intrinsic characterization of the j-crystal bases, whose original definition is artificial.

Based modules over the $\imath$quantum group of type AI

“…Proof. The assertion is a consequence of formulas ( 21) and (22). Now, we set V (ν) A := Uı A v ν , and call it the A-form of V (ν).…”

“…The aim of this paper is to define the notion of based U ı -modules without referring to crystal lattices and crystal bases, and study their properties for the ıquantum group of type AI, i.e., the ıquantum group associated with (g, k) ≃ (sl n , so n ) (see [21,22] for related works for the ıquantum group of quasi-split type AIII). By observations above, we should define a based U ı -module M to be a quadruple (M, (•, •) M , M A , ψ ı M ) consisting of a U ı -module M, a U ı -module contragredient Hermitian inner product (•, •) M , a free A-submodule of M closed under the action of Uı A , and an ıbar-involution ψ ı M , satisfying the same conditions as based U-modules.…”

Quantum symmetric pairs

“…The ıcanonical basis in split rank 1, also known as ıdivided powers [8,16], has found applications in works with Xinhong Chen and Ming Lu [20,21,50]. H. Watanabe [65,66] has developed a crystal approach (à la Kashiwara [32]) to ıcanonical bases of U ı -modules, for some quasi-split finite types.…”