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

2025/12/17 by Luc Ta, Lụ̕c Ta, Ta, Luc
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models #Advanced Topics in Algebra

paper · pdf · doi:10.1016/j.jalgebra.2026.06.036

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.

Citations

• Universal enveloping algebras of Leibniz algebras and (co)homology
• A classifying invariant of knots, the knot quandle

Related

• Yang-Baxter Solutions from Categorical Augmented Racks
• The Yang-Baxter equation, Leibniz algebras, racks and related algebraic structures
• A class of Lie racks associated to symmetric Leibniz algebras
• Racks, Leibniz algebras and Yetter-Drinfel'd modules
• From n-Leibniz algebras and linear n-racks to the solutions of the (higher analogue of) Yang-Baxter equation