From affine algebraic racks to Leibniz algebras and Yang-Baxter operators

Abstract

We introduce analogues of algebraic groups called algebraic racks, which are pointed rack objects in the category of schemes over a ground field. Addressing a problem of Loday, we construct functors assigning left and right Leibniz algebras to affine algebraic racks. These functors are compatible with closed subracks and ideals, and they recover the Lie algebras of linear algebraic groups (via conjugation quandles) and the Leibniz algebras of algebraic Lie racks. We also study properties of coordinate algebras and Leibniz algebras of affine algebraic racks. Finally, we use rack schemes to functorially construct (co-)nondegenerate Yang-Baxter operators in various categories.