The Y-Product

TL;DR

This paper introduces a topological construction that generates non-commutative Frobenius algebras, generalizes the Verlinde formula using handlebodies, and explores connections to Hopf algebra structures.

Contribution

It provides a new topological framework for non-commutative Frobenius algebras and extends the Verlinde formula to higher genus cases with a generalized S-matrix.

Findings

Constructs non-commutative Frobenius algebras from topological surfaces.

Provides a topological proof of the Verlinde formula using the solid torus.

Generalizes the Verlinde formula to higher genus handlebodies with a new S-matrix.

Abstract

We present a topological construction that provides many examples of non-commutative Frobenius algebras that generalizes the well-known pair-of-pants. When applied to the solid torus, in conjunction with Crane-Yetter theory, we provide a topological proof of the Verlinde formula. We also apply the construction to a solid handlebody of higher genus, leading to a generalization of the Verlinde formula (not the higher genus Verlinde formula); in particular, we define a generalized $S$-matrix. Finally, we discuss the relation between our construction and Yetter's construction of a handle as a Hopf algebra, and give a generalization.