# Mathematics Colloquia and Seminars

I will talk about joint work with Julia Pevtsova (arXiv:2203.10764), which concerns the small quantum group $u_q(G)$. Here $G$ is a complex reductive group and q is a root of unity in $\mathbb{C}$.We show that the category of representations for $u_q(G)$ admits a fully faithful tensor embedding into the category of coherent sheaves over a “quantum” flag variety. This quantum flag variety is, essentially, some exotic, finitely fibered space over the classical flag variety $G/B$.I will explain how this embedding theorem codifies relationships between the small quantum group and its quantum Borels, and discuss consequences in both support theory and geometric representation theory.