Decomposition of $q$-deformed Fock spaces

TL;DR

This paper decomposes level-one $q$-deformed Fock spaces of $ ext{U}_q(rak{sl}_n)$, revealing a Heisenberg algebra centralizer and an isomorphism that simplifies the structure of $q$-wedging operators.

Contribution

It introduces a novel decomposition of $q$-deformed Fock spaces, connecting them to Heisenberg algebras and tensor products of irreducible modules.

Findings

Heisenberg algebra centralizes the action of $ ext{U}_q(rak{sl}_n)$ on Fock spaces.

The $q$-deformed Fock space is isomorphic to a tensor product of a highest weight module and a Heisenberg Fock space.

Decomposition of $q$-wedging operators into components from $ ext{U}_q(rak{sl}_n)$ and the Heisenberg algebra.

Abstract

A decomposition of the level-one $q$-deformed Fock representations of $\uqn$ is given. It is found that the action of $\upqn$ on these Fock spaces is centralized by a Heisenberg algebra, which arises from the center of the affine Hecke algebra $\widehat{H}_N$ in the limit $N \rightarrow \infty$. The $q$-deformed Fock space is shown to be isomorphic as a $\upqn$-Heisenberg-bimodule to the tensor product of a level-one irreducible highest weight representation of $\upqn$ and the Fock representation of the Heisenberg algebra. The isomorphism is used to decompose the $q$-wedging operators, which are intertwiners between the $q$-deformed Fock spaces, into constituents coming from $\upqn$ and from the Heisenberg algebra.