$S_3$-permuted Frobenius Algebras

TL;DR

This paper introduces a new framework for Frobenius algebras using operads of graphs, relaxing traditional conditions and providing examples with Clifford algebras, enhancing understanding of algebraic structures.

Contribution

It presents a novel operadic approach to Frobenius algebras that drops associativity and unitality constraints, with detailed Clifford algebra examples.

Findings

Operadic graph methods effectively model Frobenius algebras.

The framework applies broadly to all Frobenius algebras.

Examples include detailed Clifford algebra constructions.

Abstract

In the present paper by Frobenius algebra Y we mean a finite dimensional algebra possessing an associative and invertible (nondegenerate) form a scalar product, referred to as the Frobenius structure. The nondegenerate form has an inverse. We drop the extra conditions of associativity and unitality of Y. Frobenius algebra is formulated within the monoidal abelian category of operad of graphs cat(m,n). Operad of graphs, i.e. diagrammatic language, is used both to illustrate the construction as well as a method of proof for the main Theorem. We give two detailed examples of this construction for Clifford algebras. Our construction, however, applies to all Frobenius algebras.