$ı$Hall algebra of the projective line and $q$-Onsager algebra

Abstract

The $ı$Hall algebra of the projective line is by definition the twisted semi-derived Ringel-Hall algebra of the category of $1$-periodic complexes of coherent sheaves on the projective line. This $ı$Hall algebra is shown to realize the universal $q$-Onsager algebra (i.e., $ı$quantum group of split affine $A_1$ type) in its Drinfeld type presentation. The $ı$Hall algebra of the Kronecker quiver was known earlier to realize the same algebra in its Serre type presentation. We then establish a derived equivalence which induces an isomorphism of these two $ı$Hall algebras, explaining the isomorphism of the $q$-Onsager algebra under the two presentations.

v2, 31 pages, minor edits, accepted by Trans. AMS

References in corpus (3)

• On triangulated orbit categories
• A Drinfeld type presentation of affine $ı$quantum groups I: split ADE type
• A conversation around the analysis of the SiP effort estimation dataset

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