Sheaf realization of Bridgeland's Hall algebra of Dynkin type

2023/03/09 by Fang, Jiepeng, Lan, Yixin, Xiao, Jie
#16G20 #17B37 #18N25 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2303.04993

Abstract

As one of results in [6], Bridgeland realized the quantum group U_v via the localization of Ringel-Hall algebra for the two-periodic projective complexes of quiver representations over a finite field. In the present paper, we generalize Lusztig's categorical construction for the nilpotent part U_v⁺ to Bridgeland's Hall algebra of Dynkin type. In particular, we obtain a basis of the Ringel-Hall algebra for the two-periodic projective complexes which has the positivity, and we categorify an integral form of the generic Bridgeland's Hall algebra which is isomorphic to the Poisson integral form of U_v, and obtain a ℤ[v,v^-1]-basis of this integral form.

Related

• Hall algebras and quantum groups associated to Dynkin quivers
• Semi-derived Hall algebras and tilting invariance of Bridgeland-Hall algebras
• Quantum Weyl Relations arising from Two-Term Complexes
• Structure coefficients for quantum groups
• The Parity of Lusztig's Restriction Functor and Green's Formula