Field and forms on $\rho$-algebras

TL;DR

This paper introduces non-commutative fields and forms on $ ho$-algebras, expanding non-commutative geometry by defining a Fr"{o}licher--Nijenhuis bracket within this new algebraic framework.

Contribution

It presents the first development of non-commutative fields, forms, and the Fr"{o}licher--Nijenhuis bracket on $ ho$-algebras, a novel class of non-commutative algebras.

Findings

Defined non-commutative fields and forms on $ ho$-algebras

Extended non-commutative geometry with a new bracket

Established foundational structures for $ ho$-algebra analysis

Abstract

In this paper we introduce non-commutative fields and forms on a new kind of non-commutative algebras: $\rho$-algebras. We also define the Fr\"{o}licher--Nijenhuis bracket in the non-commutative geometry on $\rho$-algebras.