Hall algebras and quantum symmetric pairs I: foundations

Abstract

A quantum symmetric pair consists of a quantum group $\mathbf{U}$ and its coideal subalgebra ${\mathbf{U}}_{\mathbf{\varsigma }}^{ı}$ with parameters $\mathbf{\varsigma }$ (called an $ı$quantum group). We initiate a Hall algebra approach for the categorification of $ı$quantum groups. A universal $ı$quantum group ${\tilde{\mathbf{U}}}^{ı}$ is introduced and ${\mathbf{U}}_{\mathbf{\varsigma }}^{ı}$ is recovered by a central reduction of ${\tilde{\mathbf{U}}}^{ı}$. The semi-derived 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 Appendix A by the first author. A new class of 1-Gorenstein algebras (called $ı$quiver algebras) arising from acyclic quivers with involutions is introduced. The semi-derived Ringel-Hall algebras for the Dynkin $ı$quiver algebras are shown to be isomorphic to the universal quasi-split $ı$quantum groups of finite type. Monomial bases and PBW bases for these Hall algebras and $ı$quantum groups are constructed.

v2, 74 pages, some edits and corrections, updated references, to appear in PLMS

References in corpus (5)

• On triangulated orbit categories
• Canonical bases for tensor products and super Kazhdan-Lusztig theory
• Categorification of quantum symmetric pairs I
• Semi-derived Ringel-Hall algebras and Drinfeld double
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Cited by in corpus (10)

• A Drinfeld type presentation of affine $ı$quantum groups II: split BCFG type
• $ı$Hall algebra of the projective line and $q$-Onsager algebra
• A Drinfeld type presentation of affine $ı$quantum groups I: split ADE type
• Braid group action and quasi-split affine $ı$quantum groups I
• Lectures on dualities ABC in representation theory
• Braid group symmetries on quasi-split $ı$quantum groups via $ı$Hall algebras
• $ı$Hall algebras of weighted projective lines and quantum symmetric pairs
• Serre-Lusztig relations for $ı$quantum groups III
• $ı$Hall algebra of Jordan quiver and $ı$Hall-Littlewood functions
• $ı$quantum groups of split type via derived Hall algebras