Non-involutory Hopf algebras and 3-manifold invariants

TL;DR

This paper introduces a new invariant for closed, framed 3-manifolds based on finite-dimensional Hopf algebras, extending quantum invariants but not forming a TQFT.

Contribution

It defines a novel 3-manifold invariant #(M,H) applicable to various Hopf algebra types, broadening the scope of quantum topology tools.

Findings

Invariant #(M,H) is well-defined for all finite-dimensional Hopf algebras.

When H is a quantized universal enveloping algebra, #(M,H) relates to known quantum link invariants.

The invariant does not constitute a topological quantum field theory.

Abstract

We present a definition of an invariant #(M,H), defined for every finite-dimensional Hopf algebra (or Hopf superalgebra or Hopf object) H and for every closed, framed 3-manifold M. When H is a quantized universal enveloping algebra, #(M,H) is closely related to well-known quantum link invariants such as the HOMFLY polynomial, but it is not a topological quantum field theory.