\( \imath\)Hall algebras of weighted projective lines and quantum symmetric pairs. (English) Zbl 07819594
Summary: The \(\imath\)Hall 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 \(\imath\)quantum loop algebra, which is a generalization of the \(\imath\)quantum group arising from the quantum symmetric pair of split affine type ADE in its Drinfeld type presentation. The \(\imath\)Hall algebra of the \(\imath\)quiver 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 \(\imath\)Hall algebras, explaining the isomorphism of the \(\imath\)quantum group of split affine type A under the two presentations.
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 14A22 | Noncommutative algebraic geometry |
| 16E60 | Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc. |
| 18G80 | Derived categories, triangulated categories |
Keywords:
quantum symmetric pairs; affine \(\imath\)quantum groups; Drinfeld type presentations; Hall algebras; weighted projective linesReferences:
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