2010/05/20 by Guillaume Pouchin, Pouchin, Guillaume · 1 citation
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #FOS: Mathematics #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.1005.3732
arxiv created 2010/05/20 · openalex publication_date 2010/05/20 · arxiv updated 2010/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define the Higgs algebra H_\P1 of the projective line, as a convolution algebra of constructible functions on the global nilpotent cone \underlineΛ_\P1, a lagrangian substack of the Higgs bundle T^*\Coh_\P1, where \Coh_\P1 is the stack of coherent sheaves on \P1. We prove that H_\P1 is isomorphic to (some completion of) U+(sl2). We use this geometric realization to define a semicanonical basis of U+(sl2), indexed by irreducible components of \underlineΛ_\P1. We also construct a combinatorial data on this set of irreducible components in the spirit of \citeKS, which is an affine analog of a crystal. We call it a loop crystal and give some of its properties.