$\imath$Hall algebras of weighted projective lines and quantum symmetric pairs

Abstract

The ıHall algebra of a weighted projective line is defined to be the semi-derived Ringel-Hall algebra of the category of 1-periodic complexes of coherent sheaves on the weighted projective line over a finite field. We show that this Hall algebra provides a realization of the ıquantum loop algebra, which is a generalization of the ıquantum group arising from the quantum symmetric pair of split affine type ADE in its Drinfeld type presentation. The ıHall algebra of the ıquiver algebra of split affine type A was known earlier to realize the same algebra in its Serre presentation. We then establish a derived equivalence which induces an isomorphism of these two ıHall algebras, explaining the isomorphism of the ıquantum group of split affine type A under the two presentations.

Hall algebras and quantum symmetric pairs III: Quiver varieties

“…Let C n (n ≥ 2) be the oriented cyclic quiver with n vertices. With the help of Theorem 5.1, we can use the ıHall algebra of C n with trivial involution to realize the ıquantum groups of sl n and gl n , especially for the case n = 2 where (C 2 , Id) is not a virtually acyclic ıquiver; see [13,Section 10].…”

Hall Algebras and Quantum Symmetric Pairs II: Reflection Functors

“…ıQuantum loop algebras are a further generalization of ıquantum groups of split affine ADE type considered in [LW21b]. In [LR21], we established a homomorphism Ω from the ıquantum loop algebra Dr U ı (Lg) of split type to the ıHall algebra of the weighted projective line ı H(X k ). This is a sequel of [LR21] which is devoted to proving the morphism Ω : Dr U ı (Lg) → ı H(X k ) given in [LR21] is injective if g is of finite or affine type.…”

“…In [LR21], we established a homomorphism Ω from the ıquantum loop algebra Dr U ı (Lg) of split type to the ıHall algebra of the weighted projective line ı H(X k ). This is a sequel of [LR21] which is devoted to proving the morphism Ω : Dr U ı (Lg) → ı H(X k ) given in [LR21] is injective if g is of finite or affine type. The idea of the proof is to construct the inverse of Ω by giving a presentation of the composition subalgebra ı C(X k ) of ı H(X k ).…”

“…In Section 2, we recall the necessary materials on ıquantum groups, Hall algebras and ıHall algebras, weighted projective lines, etc. In Section 3, we recall the construction of the morphism Ω given in [LR21], and formulate the main result. Section 4 is devoted to realizing the ıquantum loop algebra of affine gl n via the ıHall algebra of the cyclic quiver C n .…”

Quantum symmetric pairs

“…This provides an ı-analog of (7). Lu and Ruan are developing ıHall algebras of the weighted projective lines [45] to realize the affine ıquantum groups in the new presentation, generalizing the rank one construction in [46].…”