From affine algebraic racks to Leibniz algebras

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 a functor assigning a canonical Leibniz algebra to every affine algebraic rack. This functor recovers the Lie algebras of affine algebraic groups (via conjugation quandles) and the Leibniz algebras of algebraic Lie racks, and it is compatible with closed subracks and left ideals. We also compare properties of affine algebraic racks, commutative corack algebras, and their Leibniz algebras.