Algebraic Structures Derived from Foams

TL;DR

This paper explores algebraic structures such as Lie brackets and bialgebras derived from foam surfaces in TQFT, connecting these to Frobenius algebras and their diagrammatic representations.

Contribution

It introduces new algebraic operations from foam branch lines and analyzes their relations to Frobenius algebra structures in TQFT.

Findings

Lie bracket structures are derived from foam branch lines.

Bialgebra structures are constructed and related to Frobenius algebras.

Diagrammatic relations illustrate the connections between these algebraic structures.

Abstract

Foams are surfaces with branch lines at which three sheets merge. They have been used in the categorification of sl(3) quantum knot invariants and also in physics. The 2D-TQFT of surfaces, on the other hand, is classified by means of commutative Frobenius algebras, where saddle points correspond to multiplication and comultiplication. In this paper, we explore algebraic operations that branch lines derive under TQFT. In particular, we investigate Lie bracket and bialgebra structures. Relations to the original Frobenius algebra structures are discussed both algebraically and diagrammatically.