Polynomial Relations in the Centre of U_q(sl(N))

TL;DR

This paper studies the polynomial relations among central elements of the quantum group U_q(sl(N)) at roots of unity, providing explicit formulas and exploring implications for representations and fusion rules.

Contribution

It explicitly derives polynomial relations among the central generators of U_q(sl(N)) at roots of unity, generalizing known results from U_q(sl(2)).

Findings

Explicit polynomial relations among central generators are obtained.

Application to parametrization of irreducible representations is discussed.

Implications for fusion rules are sketched.

Abstract

When the parameter of deformation q is a m-th root of unity, the centre of U_q(sl(N))$ contains, besides the usual q-deformed Casimirs, a set of new generators, which are basically the m-th powers of all the Cartan generators of U_q(sl(N)). All these central elements are however not independent. In this letter, generalising the well-known case of U_q(sl(2)), we explicitly write polynomial relations satisfied by the generators of the centre. Application to the parametrization of irreducible representations and to fusion rules are sketched.