Affine Supertrusses and Superbraces

TL;DR

This paper introduces affine supertrusses and superbraces, extending algebraic structures like trusses and braces into the supermathematics setting, and explores their applications to the Yang--Baxter equation.

Contribution

It defines affine supertrusses and superbraces, introduces the concept of cotrusses, and generalizes Rump's braces and the Yang--Baxter equation to superalgebraic contexts.

Findings

Constructed affine superbrace from affine supertruss.

Generalized Rump's braces to supermathematics.

Proposed a superalgebraic version of the Yang--Baxter equation.

Abstract

Brzezi\'nski's trusses are ``ring-like'' algebraic structures in which the addition is replaced with an abelian heap operation and the binary product satisfies a natural distributivity rule of the ternary product. The question of how to define ($\mathbb{Z}_2$-graded) super-versions of trusses is addressed in this note. Taking our cue from the theory of algebraic supergroups, we define an affine supertruss as a representable functor from the category of unital associative supercommutative superalgebras to the category of trusses. The representing superalgebras are equipped with a `cotruss' structure--a new concept in itself. We show that from an affine supertruss one can construct an affine superbrace, and so generalise Rump's braces to supermathematics. As an application of these constructions, we propose a generalisation of the set-theoretic Yang--Baxter equation to the setting of affine superschemes.