| Budget Amount *help |
¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
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| Outline of Final Research Achievements |
The project fits into the realm of geometric representation theory, which stems from synergies between algebra and geometry. The goal of the project was the discovery of new algebraic structures (quantum groups) arising from the study of (moduli) spaces in geometry. We have introduced new quantum groups naturally associated to curves (opposite to those known in the literature, associated to quivers). Our approach to their definition has been geometric, based on the theory of Hall algebras and their refined versions (cohomological, K-theoretical, etc). Specifically, we have defined three algebras: the Betti, the de Rham, and the Dolbeault algebras of a curve. They represent new symmetries arising from the geometry of the corresponding moduli spaces, which play a preeminent role in geometry (e.g., in non-abelian Hodge theory) and in physics (e.g., in gauge theory). Thus, they unlock a new striking connection between geometry, algebra, and physics, which needs to be investigated further.
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