From affine algebraic racks to Leibniz algebras
Luc Ta
TL;DR
The paper introduces affine algebraic racks and proves a canonical functor that assigns to each rack Q=Spec(A) a Leibniz algebra 𝔮=Der_k(A,k) with bracket [D,E]=(D⊗E)∘∇, unifying a Lie-algebra picture for conjugation racks with Leibniz-algebras arising from Lie racks. This construction recovers the Lie algebras of affine algebraic groups via Conj(G) and yields Leibniz algebras for algebraic Lie racks, while remaining compatible with subracks and left ideals. The work develops a robust algebraic framework of racks and corack algebras, provides explicit examples (e.g., OL_n) of non-Lie Leibniz algebras, and demonstrates strong functorial and structural properties linking rack geometry to Leibniz theory. It also charts a path for further study of representations, cohomology, and extensions of algebraic racks beyond the affine setting.
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.