Graded Frobenius Algebras

TL;DR

This paper introduces a PROP framework for graded Frobenius algebras, connecting topological quantum field theories, cohomology, and homology, with detailed sign analysis in algebra suspensions.

Contribution

It constructs a PROP encoding 2D-TQFTs with grading and characterizes graded Frobenius algebras via maps and relations, linking algebraic and topological structures.

Findings

Defines a graded Frobenius algebra as an algebra over a new PROP.

Connects graded Frobenius algebras to cohomology, loop homology, and Hochschild homology.

Provides a detailed analysis of signs in suspending algebras over PROPs.

Abstract

We construct a PROP which encodes 2D-TQFTs with a grading. This defines a graded Frobenius algebra as algebras over this PROP. We also give a description of graded Frobenius algebras in terms of maps and relations. This structure naturally arises as the cohomology of manifolds, loop homology and Hochschild homology of Frobenius algebras. In addition, we give a comprehensive description of the signs that arise in suspending algebras over PROPs.