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

TL;DR

This paper introduces algebraic racks as analogues of algebraic groups, constructs functors to Leibniz algebras, and uses rack schemes to build Yang-Baxter operators, connecting algebraic geometry with algebraic structures.

Contribution

It defines algebraic racks in schemes, constructs functors to Leibniz algebras, and develops methods to produce Yang-Baxter operators from rack schemes.

Findings

Algebraic racks generalize algebraic groups.

Functors from racks to Leibniz algebras are established.

Rack schemes can produce 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.