Hall algebras and quantum symmetric pairs I: Foundations

Abstract

A quantum symmetric pair consists of a quantum group 𝐔 and its coideal subalgebra 𝐔 𝚤 𝝇 with parameters 𝝇 (called an 𝚤quantum group). We initiate a Hall algebra approach for the categorification of 𝚤quantum groups. A universal 𝚤quantum group Ũ𝚤 is introduced and 𝐔 𝚤 𝝇 is recovered by a central reduction of Ũ𝚤 . The semiderived Ringel-Hall algebras of the first author and Peng, which are closely related to semi-derived Hall algebras of Gorsky and motivated by Bridgeland's work, are extended to the setting of 1-Gorenstein algebras, as shown in the Appendix by the first author. A new class of 1-Gorenstein algebras (called 𝚤quiver algebras) arising from acyclic quivers with involutions is introduced. The semi-derived Ringel-Hall algebras for the Dynkin 𝚤quiver algebras are shown to be isomorphic to the universal quasi-split 𝚤quantum groups of finite type. Monomial bases and PBW bases for these Hall algebras and 𝚤quantum groups are constructed.

Quantum Borcherds–Bozec algebras via semi‐derived Ringel–Hall algebras II: Braid group actions

“…Proof The $$\imath$$quiver algebra and its module category are considered in [17], and the category $${\mathcal {C}}_{{\mathbb {Z}}_2}(\operatorname{rep}\nolimits _\mathbf {k}^{\rm nil}(Q))$$ can be viewed to be the module category of an $$\imath$$quiver algebra of diagonal type; see [17, Example 2.10]. So, the statement follows from [18, Theorem 2.11].$$\Box$$…”

Braid group action and quasi-split affine $\imath$quantum groups I

“…The ıHall algebra constructions based on ıquivers or ıweighted projective lines (cf. [LW22,LR21]) can be used to realize all affine quasi-split ıquantum groups except A 2r with nontrivial diagram involution. The simplest affine case not covered by the current ıHall algebra approach is exactly U ı ( sl 3 , τ ).…”

“…Recall I = {0, 1, 2}. Let U ı := U ı ( g) be the universal ıquantum group associated to the Satake diagram (2.3); see [LW22,CLW21] (also cf. [Let99,Ko14]).…”

“…The ıquantum groups U ı arising from quantum symmetric pairs (U, U ı ) associated to Satake diagrams [Let99] (see [Ko14]) can be viewed as a vast generalization of Drinfeld-Jimbo quantum groups associated to Dynkin diagrams; see the survey [W22] and references therein. In this paper, we shall work with universal ıquantum groups ( U, U ı ) following [LW22,WZ22], where U is the Drinfeld double quantum group, as this allows us to formulate the relative braid group action conceptually. Just as the quantum group U is obtained by U by a central reduction, a central reduction from universal ıquantum groups recovers ıquantum groups with parameters [Let02,Ko14].…”

A Geometric Realization of Symmetric Pairs of Type AIII

“…It is worth to mention that there are other geometric ways to realize quantum symmetric pairs, for example [LW21]. It is based on the Hall algebra construction [LW19].…”