A description of continuous rigid motion compatible Minkowski valuations is established. As an application, we present a Brunn-Minkowski type inequality for intrinsic volumes of these valuations

In this article we prove that if $G$ is a connected, simply connected, semisimple algebraic group over an algebraically closed field of sufficiently large characteristic, then all the blocks of the restricted enveloping algebra $(U\mathfrak{g}{)}_0$ of the Lie algebra $\mathfrak{g}$ of $G$ can be endowed with a Koszul grading (extending results of Andersen, Jantzen, and Soergel). We also give information about the Koszul dual rings. In the case of the block associated to a regular character $\lambda$ of the Harish-Chandra center, the dual ring is related to modules over the specialized algebra $(U\mathfrak{g}{)}^{\lambda }$ with generalized trivial Frobenius character. Our main tool is the localization theory developed by Bezrukavnikov, Mirković, and Rumynin

We introduce the concept of a geometric categorical ${\mathfrak{sl}}_2$ action and relate it to that of a strong categorical ${\mathfrak{sl}}_2$ action. The latter is a special kind of $2$-representation in the sense of Lauda and Rouquier. The main result is that a geometric categorical ${\mathfrak{sl}}_2$ action induces a strong categorical ${\mathfrak{sl}}_2$ action. This allows one to apply the theory of strong ${\mathfrak{sl}}_2$ actions to various geometric situations. Our main example is the construction of a geometric categorical ${\mathfrak{sl}}_2$ action on the derived category of coherent sheaves on cotangent bundles of Grassmannians

Lusztig constructed a Frobenius morphism for quantum groups at an $\ell$th root of unity, which gives an integral lift of the Frobenius map on universal enveloping algebras in positive characteristic. Using the Hall algebra, we give a simple construction of this map for the positive part of the quantum group attached to an arbitrary Cartan datum in the nondivisible case