Glueing operation for r-matrices, quantum groups and link-invariants of Hecke type

TL;DR

This paper introduces a new associative glueing operation for solutions of the Quantum Yang-Baxter Equation of Hecke type, extending to quantum groups, vector spaces, and link-invariants, enabling systematic construction of complex invariants.

Contribution

It presents a novel glueing operation for R-matrices, quantum groups, and link invariants, facilitating the construction of complex structures from simpler ones.

Findings

Defined an associative glueing operation for R-matrices of Hecke type.

Extended the operation to quantum groups and quantum vector spaces.

Constructed link-invariants using a state-sum model with Boltzmann weights.

Abstract

We introduce an associative glueing operation $\oplus_q$ on the space of solutions of the Quantum Yang-Baxter Equations of Hecke type. The corresponding glueing operations for the associated quantum groups and quantum vector spaces are also found. The former involves $2\times 2$ quantum matrices whose entries are themselves square or rectangular quantum matrices. The corresponding glueing operation for link-invariants is introduced and involves a state-sum model with Boltzmann weights determined by the link invariants to be glued. The standard $su(n)$ solution, its associated quantum matrix group, quantum space and link-invariant arise at once by repeated glueing of the one-dimensional case.