Integral Forms of Hall Algebras and their Limits

In this thesis we tell the story of how two isomorphic algebras - quantized enveloping algebras and Bridgeland-Hall algebras - are simultaneous deformations of two simpler algebras: the universal enveloping algebra of a Lie algebra and the coordinate algebra of a Poisson-Lie group. We will also explain how a similar deformation picture holds for Hall algebras, of which Bridgeland-Hall algebras are a generalization, and a subalgebra of the quantized enveloping algebra called its positive part. Our particular contribution to this story is to establish the precise way in which Bridgeland-Hall algebras deform coordinate algebras of Poisson-Lie groups. We will give a calculation of the Hall algebraic structure of the resulting Poisson-Lie groups and also explain the relationship with how quantized enveloping algebras deform coordinate algebras of Poisson-Lie groups. Using the Bridgeland-Hall algebra approach to Poisson-Lie groups we will give a new way to extract simple Lie algebras from Bridgeland-Hall algebras and in addition provide a computation of the Hall algebraic structure of these Lie algebras. Finally we provide a new, more direct proof of an old but tricky to prove theorem due to De Concini and Procesi that quantized enveloping algebras are deformations of the coordinate algebra of a particular Poisson-Lie group called the standard dual Poisson-Lie group.