Faculty of Mathematics
Collaborative Research Centre 701
Spectral Structures and Topological Methods in Mathematics

# Friday, April 21, 2017 - 14:30 in H10

## Applications of mutations in the derived categories of weighted projective lines to Lie and quantum algebras

A talk in the 'Seminar Representation Theory of Algebras' series by
Shiquan Ruan from Beijing
 Abstract: Let $\text{coh-}X$ be the category of coherent sheaves over a weighted projective line $X$ and let $D^b(\text{coh-}X)$ be its bounded derived category. In this talk we will focus on the study of the right and left mutation functors arising in $D^b(\text{coh-}X)$ attached to certain line bundles. We first show that these mutation functors give rise to simple reflections for the Weyl group of the star shaped quiver $Q$ associated with $X$. By further dealing with the Ringel-Hall algebra of $X$, we show that these functors provide a realization for Tits’ automorphisms of the Kac-Moody algebra $g_Q$ of $Q$, as well as for Lusztig’s symmetries of the quantum enveloping algebra of $g_Q$.