𝚤Hall algebra of the projective line and 𝑞-Onsager algebra

Abstract

Theı\imathHall algebra of the projective line is by definition the twisted semi-derived Ringel-Hall algebra of the category of11-periodic complexes of coherent sheaves on the projective line. Thisı\imathHall algebra is shown to realize the universalqq-Onsager algebra (i.e.,ı\imathquantum group of split affineA1A_1type) in its Drinfeld type presentation. Theı\imathHall 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 twoı\imathHall algebras, explaining the isomorphism of theqq-Onsager algebra under the two presentations.