| 研究実績の概要 |
FY2019 has been devoted to the study of Dolbeault cohomological Hall algebra of the projective line. It is expected that this algebra is isomorphic to the cohomological Hall algebra of the preprojective algebra of the quiver A_1^{1}, which is (a positive part of) the affine Yangian of type A_1^{1}.
This is geometrically justified by the derived McKay correspondence for the type A_1 singularity. The Dolbeault algebra should provide a half of the affine Yangian which is different from the half provided by the quiver construction, since the above derived equivalence is not t-exact.
The approach pursued in FY2019 was the study of the relationship between the Dolbeault algebra and the mentioned derived equivalence. Our first result in such a direction is the following one.
In [arXiv:2004.13685] (which was posted only in April 2020 because of the confinement due to COVID-19 Pandemic in Italy, France, and USA), the semistable Dolbeault algebra is described explicitly (and more generally I obtained a characterization of the semistable cohomological Hall algebra of the resolution of the A_N singularity).
The understanding of the full Dolbeault algebra is a subject of a current investigation: I am investigating the relation between the cohomological Hall algebra of the resolution of the A_N singularity (for N=1, one has the Dolbeault algebra) and the braid group action coming from the derived category of sheaves on the resolution of the A_N singularity. (This approach is analogous to that of Beck for quantum enveloping algebras.)
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