One-Dimensional Anyon Lattice and sl_q(2) Algebra Realization

We discuss the connection between anyons (particles with fractional statistics) living on a one-dimensional lattice and sl_q(2) algebra. We assign to each site-anyon field, then we demand the anyonic fields to be noncommuting objects in agreement with the Chern--Simons picture of anyons. We show how the anyonic algebra can emerge from these noncommuting objects. Starting from the emerged algebra we build the sl_q(2) algebra at q roots of unity.