Quantum matrix algebra for the SU(n) WZNW model

TL;DR

This paper explores the quantum matrix algebra arising from the SU(n) WZNW model's zero modes, demonstrating its connection to the quantum universal enveloping algebra U_q(sl_n) and analyzing its structure at roots of unity.

Contribution

It generalizes the Fock space representation of the algebra for generic q and characterizes its finite-dimensional quotient at roots of unity, linking to quantum groups.

Findings

Fock space representation yields each irreducible U_q(sl_n) with multiplicity one.

At roots of unity, the algebra admits a finite-dimensional quotient.

The algebra structure relates to the quantum universal enveloping algebra.

Abstract

The zero modes of the chiral SU(n) WZNW model give rise to an intertwining quantum matrix algebra A generated by an n x n matrix a=(a^i_\alpha) (with noncommuting entries) and by rational functions of n commuting elements q^{p_i}. We study a generalization of the Fock space (F) representation of A for generic q (q not a root of unity) and demonstrate that it gives rise to a model of the quantum universal enveloping algebra U_q(sl_n), each irreducible representation entering F with multiplicity 1. For an integer level k the complex parameter q is an even root of unity, q^h=-1 (h=k+n) and the algebra A has an ideal I_h such that the factor algebra A_h = A/I_h is finite dimensional.