Canonical bases arising from quantum symmetric pairs

Abstract

For quantum symmetric pairs (U, U ı ) of Kac-Moody type, we construct ıcanonical bases for the highest weight integrable U-modules and their tensor products regarded as U ı -modules, as well as an ıcanonical basis for the modified form of the ıquantum group U ı . A key new ingredient is a family of explicit elements called ıdivided powers, which are shown to generate the integral form ofU ı . We prove a conjecture of Balagovic-Kolb, removing a major technical assumption in the theory of quantum symmetric pairs. Even for quantum symmetric pairs of finite type, our new approach simplifies and strengthens the integrality of quasi-K-matrix and the constructions of ıcanonical bases, by avoiding a case-by-case rank one analysis and removing the strong constraints on the parameters in a previous work.

Kazhdan-Lusztig theory of super type D and quantum symmetric pairs

“…They established the commutativity between the actions of U [BW13] applies to the coideal subalgebras with different parameters without difficulty, that is, simple U q (sl k )-modules and their tensor products admit ı-canonical bases. In the ongoing work [BW16], we generalize the construction of ı-canonical bases to more general quantum symmetric pairs (see also Remark 2.17).…”

“…Section 2.1 -Section 2.3 are analogous to [BW13, Part 1] hence we shall omit the proofs almost entirely. Some of the quantum symmetric pairs considered here are of different parameters from the ones considered in [BW13] (see also [BW16] for more general construction). We refer to [Ko] for the general theory of quantum symmetric pairs.…”

Positivity VS Negativity of Canonical Bases

“…There is another purely representation theoretic approach in [4] toward the bilinear forms and canonical bases for general quantum coideal algebras including iU (sl n ), which nevertheless cannot address the positivity of canonical bases. Note that the papers [16,17,20,18] are mostly concerned about the quantum Schur algebras and quantum groups of affine type A.…”

Schur Algebras and Quantum Symmetric Pairs With Unequal Parameters

“…U ı ) at the specialization u " v Lps 0 q , v " v Lps 1 q . A general theory of canonical bases for quantum symmetric pairs with parameters of arbitrary finite type was developed in [BW16].…”