Cluster Structures on Double Bott–Samelson Cells

Abstract

Let $\mathsf {C}$ be a symmetrisable generalised Cartan matrix. We introduce four different versions of double Bott–Samelson cells for every pair of positive braids in the generalised braid group associated to $\mathsf {C}$ . We prove that the decorated double Bott–Samelson cells are smooth affine varieties, whose coordinate rings are naturally isomorphic to upper cluster algebras.

Keywords

• cluster varieties
• double Bott-Samelson cells
• Donaldson-Thomas transformation
• Legendrian links
• 13F60
• 14M15
• 14N35
• 57K14