On an analogue of Frobenius formalism for 3-algebras and pentagon equations solutions arising from projectors

TL;DR

This paper introduces a Frobenius-like compatibility for 3-algebras, enabling the construction of full 3-algebras and invariants for lens spaces, along with new solutions to pentagon equations derived from projectors.

Contribution

It extends Ruth Lawrence's 3-algebra framework by proposing a Frobenius analogue, providing explicit examples and invariants, and discovering new pentagon equation solutions from projectors.

Findings

Constructed a full 3-algebra using Frobenius compatibility.

Developed invariants for lens spaces based on new 3-algebras.

Discovered a new family of pentagon equations solutions from projectors.

Abstract

Ruth J.Lawrence introduced a notion of a 3-algebra to construct invariants of 3-manifolds based on their triangulations in her paper "Algebras and triangular relations". Her primary definition is suitable for certain triangulations only although a hint to handle arbitrary ones has been proposed. Here I introduce an analogue of Frobenius compatibility for one class of 3-algebras hence I obtain a way to construct a full 3-algebra. Additionally, I provide with examples of a 3-algebra and invariants for lens spaces. Moreover, it leads to a new family of pentagon equations solutions: arising from projectors.