Flip-Moves and Graded Associative Algebras

TL;DR

This paper explores the connection between flip-moves in polygonal decompositions and graded associative algebras, extending previous work on topological field theories and triangulations.

Contribution

It introduces a broader relation between flip-moves involving n-gons and Z_{n-2}-graded associative algebras, generalizing earlier results.

Findings

Flip-invariant models can be constructed from Z_{2} or Z_{1}-graded algebras.

The approach generalizes the graphical interpretation of associativity.

Connections to three-dimensional topological lattice theories are discussed.

Abstract

The relation between discrete topological field theories on triangulations of two-dimensional manifolds and associative algebras was worked out recently. The starting point for this development was the graphical interpretation of the associativity as flip of triangles. We show that there is a more general relation between flip-moves with two $n$-gons and $Z_{n-2}$-graded associative algebras. A detailed examination shows that flip-invariant models on a lattice of $n$-gons can be constructed {}from $Z_{2}$- or $Z_{1}$-graded algebras, reducing in the second case to triangulations of the two-dimensional manifolds. Related problems occure naturally in three-dimensional topological lattice theories.