Lusztig sheaves and tensor products of integrable highest weight modules

Abstract: By introducing $N$-framed quivers, we define the localization of Lusztig's sheaves for $N$-framed quivers and functors $E^{(n)}_{i}, F^{(n)}_{i}, K^{\pm}_i$ for localizations. This gives a categorical realization of tensor products of integrable highest weight modules of the quantized enveloping algebra. The simple perverse sheaves in the localization provide a basis of the tensor product. We prove that this basis coincides with the canonical basis of tensor product in the sense of Lusztig and Bao-Wang. Moreover, we give a categorical interpretation of the Yang-Baxter equation.