iHALL ALGEBRA OF THE PROJECTIVE LINE AND q-ONSAGER ALGEBRA

The iHall 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 iHall algebra is shown to realize the universal q-Onsager algebra (i.e., iquantum group of split affine A1 type) in its Drinfeld type presentation. The iHall 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 iHall algebras, explaining the isomorphism of the q-Onsager algebra under the two presentations.