Bosonization of Coordinate Ring of $U_q(SL(N))$. The Cases of $N=2$ and $N=3$

TL;DR

This paper explicitly constructs representations of the non-abelian coordinate ring of quantum groups $U_q(SL(N))$ for $N=2,3$ using creation and annihilation operators, providing concrete realizations of these algebraic structures.

Contribution

It offers explicit bosonization of the coordinate ring of $U_q(SL(N))$ for $N=2,3$, making their representations more accessible and concrete.

Findings

Explicit representations for $N=2,3$ constructed

Representation formulas using creation and annihilation operators provided

Potential for generalization to higher $N$ and Kac-Moody algebras suggested

Abstract

Non-abelian coordinate ring of $U_q(SL(N))$ (quantum deformation of the algebra of functions) for $N=2,3$ is represented in terms of conventional creation and annihilation operators. This allows to construct explicitly representations of this algebra, which were earlier described in somewhat more abstract algebraic fashion. Generalizations to $N>3$ and Kac-Moody algebras are not discussed but look straightforward.