Quantum supergroups and topological invariants of three - manifolds

TL;DR

This paper extends the Reshetikhin-Turaev method to quantum supergroups, developing a general approach for constructing three-manifold invariants using eigenvalues of central elements, demonstrated with $U_q(gl(2|1))$ at roots of unity.

Contribution

It introduces a novel method for three-manifold invariants based on quantum supergroups, broadening the scope of topological invariants in quantum topology.

Findings

Constructed new three-manifold invariants using quantum supergroups.

Applied the method to $U_q(gl(2|1))$ at odd roots of unity.

Demonstrated the effectiveness of the approach with explicit examples.

Abstract

The Reshetikhin - Turaeve approach to topological invariants of three - manifolds is generalized to quantum supergroups. A general method for constructing three - manifold invariants is developed, which requires only the study of the eigenvalues of certain central elements of the quantum supergroup in irreducible representations. To illustrate how the method works, $U_q(gl(2|1))$ at odd roots of unity is studied in detail, and the corresponding topological invariants are obtained.