Quantum SUSY operads

TL;DR

This paper extends the concept of operads to include supersymmetric (fermionic) quantum structures, analyzing their algebraic properties and symmetries, and connecting them to graphical and coding frameworks.

Contribution

It introduces a framework for lifting supersymmetric operads to quantum operads and explores the operad structure and symmetries of 1D SUSY algebras.

Findings

Operads of supersymmetric rings are constructed.

1D SUSY algebras possess an operad structure.

Connections to Adinkra graphs, dessins, and codes are established.

Abstract

In a recent paper, we described a lifting of coordinate rings of groups, loops, quantum groups, etc. to the categoric setup of operads. In most examples of that paper, these rings are non--commutative. Quantum physics of the XX--th century added one more, quite nontrivial degree of freedom: coordinates might become fermionic. In their classical version, the fermionic coordinates anti--commute, and the resulting rings are called supersymmetric, or SUSY, ones. In this paper, we try to lift operads involving fermionic coordinates to quantum operads. We have to restrict ourselves by lifting operads of supersymmetric rings. We also show that $1D$ supersymmetric algebras have an operad structure, and we analyze their symmetries, through their relation to Adinkra graphs, dessins and codes.