| Abstract: |
| There are two important ways to realize quantum groups, one is Hall algebra given by Ringel and Bridgeland, the other one is the convolution algebra of perverse sheaves given by Lusztig, Nakajima and Qin. In this talk, we shall compare these two realizations. The perverse sheaves give the dual canonical basis of quantum groups by Hernandez-Leclerc and Qin. We prove that dual canonical basis is invariant under braid group actions, and the transition matrix from the dual canonical basis to the basis of Hall algebra is integral and positive, which extend Lusztig`s result. We also compute the rank 1 dual canonical bases, which coincide with the double canonical bases defined by Bernstein and Greenstein, so we expect that there two bases coincide. This is joint work with Xiaolong Pan. |
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