[Paper Review] Decomposition of $q$-deformed Fock spaces

Masaki Kashiwara, Tetsuji Miwa+1 more|ArXiv.org|Aug 14, 1995

Algebraic structures and combinatorial models4 references20 citations

TL;DR

This paper establishes a decomposition of the level-one $q$-deformed Fock space for $U'_q(\widehat{\mathfrak{sl}}_n)$ as a $U'_q(\widehat{\mathfrak{sl}}_n)$-Heisenberg bimodule, showing it is isomorphic to the tensor product of a highest-weight representation of $U'_q(\widehat{\mathfrak{sl}}_n)$ and the Fock module of a Heisenberg algebra. The key result is that $q$-wedge operators decompose into components from the quantum affine algebra and the Heisenberg algebra, with explicit realization via vertex operators and commutation relations.

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 $N ightarrow \infty$. 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.

Motivation & Objective

• To decompose the level-one $q$-deformed Fock space of $U'_q(\widehat{\mathfrak{sl}}_n)$ into irreducible components.
• To identify the central action of a Heisenberg algebra on the $q$-deformed Fock space, arising as a limit of the center of the affine Hecke algebra $\widehat{H}_N$ as $N \to \infty$.
• To realize $q$-wedge operators as intertwiners and decompose them into contributions from $U'_q(\widehat{\mathfrak{sl}}_n)$ and the Heisenberg algebra.
• To establish a bimodule isomorphism between the $q$-deformed Fock space and the tensor product of a highest-weight representation and the Heisenberg Fock module.

Proposed method

• Construct the $q$-deformed Fock space via infinite $q$-wedges on a tensor product of evaluation modules $V(z) = \mathbb{C}^n \otimes \mathbb{C}[z,z^{-1}]$.
• Define the action of $U'_q(\widehat{\mathfrak{sl}}_n)$ on $V(z)$ using generators $E_i, F_i, K_i^{\pm 1}$ with $q$-Serre relations and $q$-deformed commutation rules.
• Introduce a Heisenberg algebra generated by operators $B_a$ acting on the Fock space, with commutator $[B_a, B_{-a}] = a \frac{1 - q^{2na}}{1 - q^{2a}}$.
• Realize the $q$-wedge intertwiner $\Omega'(w)$ as a tensor product of a $U'_q(\widehat{\mathfrak{sl}}_n)$-intertwiner $\tilde{\Phi}^*(w)$ and a vertex operator $\Xi(w)$ for the Heisenberg algebra.
• Use two-point functions of vertex operators to compare matrix elements and derive the commutator $[B_a, B_{-a}]$ via coefficient matching in generating series.
• Leverage the Hopf algebra structure of $U_q(\widehat{\mathfrak{sl}}_n)$ and the $d$-grading to preserve degree in the intertwiner construction.

Experimental results

Research questions

• RQ1 How does the action of $U'_q(\widehat{\mathfrak{sl}}_n)$ on the $q$-deformed Fock space decompose when a Heisenberg algebra acts centrally?
• RQ2 What is the precise structure of the $q$-wedge intertwiners between $q$-deformed Fock spaces?
• RQ3 Can the $q$-deformed Fock space be realized as a bimodule over $U'_q(\widehat{\mathfrak{sl}}_n)$ and a Heisenberg algebra?
• RQ4 What is the commutation relation between the Heisenberg generators $B_a$ and $B_{-a}$ that arises from the $q$-wedge intertwiner?
• RQ5 How do the two-point functions of the $q$-wedge operators constrain the structure of the Heisenberg algebra?

Key findings

• The $q$-deformed Fock space is isomorphic as a $U'_q(\widehat{\mathfrak{sl}}_n)$-Heisenberg bimodule to the tensor product of a level-one irreducible highest-weight representation of $U'_q(\widehat{\mathfrak{sl}}_n)$ and the Fock module of the Heisenberg algebra.
• The $q$-wedge intertwiner $\Omega'(w)$ decomposes as $\tilde{\Phi}^*(w) \otimes \Xi(w)$, where $\tilde{\Phi}^*(w)$ is a $U'_q(\widehat{\mathfrak{sl}}_n)$-intertwiner and $\Xi(w)$ is a vertex operator for the Heisenberg algebra.
• The commutator $[B_a, B_{-a}]$ is explicitly computed as $a \frac{1 - q^{2na}}{1 - q^{2a}}$ via comparison of two-point functions of vertex operators.
• The two-point function of the $q$-wedge operator $\Omega_m(w)$ is given by $\frac{1 - w_2/w_1}{1 - q^2 w_2/w_1}$, which matches the product of the $U'_q(\widehat{\mathfrak{sl}}_n)$ and Heisenberg contributions.
• The vertex operator $\Xi(w)$ is realized as $\exp\left(\sum_{b \geq 1} \frac{B_{-b} w^b}{\gamma_b}\right) \exp\left(-\sum_{b \geq 1} \frac{B_b w^{-b}}{\gamma_b}\right)$ with $\gamma_b = [B_b, B_{-b}]$.
• The isomorphism is preserved under the action of both $U'_q(\widehat{\mathfrak{sl}}_n)$ and the Heisenberg algebra, confirming the bimodule structure.