Anyonic Quantum Groups

Abstract

We introduce non-standard quantum group structures on the finite groups ℤ n . These determine non-trivial braidings Ψ in the category of ℤ n -graded vector spaces. The braiding is an anyonic one, \( \Psi (\nu \, \otimes \,w)\, = \,{e^{{{2{\pi _z}\left| {\left. v \right\|w} \right|} \over n}}}w\, \otimes v \) for homogeneous elements of degree |v|, |w|. This category of anyonic vector spaces generalizes that of super vector spaces, which are recovered as n = 2. We give examples of anyonic quantum groups. These are like super quantum groups with ±1 statistics generalised to anyonic ones. They include examples obtained by transmutation of u q [sl 2) at a root of unity.