Réamonn Ó Buachalla
Levi-Civita Conections for the Irreducible Quantum Flag Manifolds
In the 2000s a series of seminal papers by Heckenberger and Kolb introduced an essentially unique covariant $q$-deformed de Rham complex for the irreducible quantum flag manifolds. In the years since, it has become increasingly clear that these differential graded algebras have a central role to play in understanding the noncommutative geometry of the Drinfeld–Jimbo quantum groups. In this talk we present the recent classification of covariant Levi-Civita metrics (in the sense of Beggs and Majid) for these differential calculi. Moreover, we will show how the bimodule map of these connections allows us to understand the Heckenberger-Kolb calculi as quantum exterior algebras. Time permitting, we will discuss the extension of this work to the full quantum flag manifolds.
