[Paper Review] Bosonic Realizations of $U_q(C^{(1)}_n)$

Naihuan Jing, Yoshitaka Koyama+1 more|ArXiv.org|Jan 28, 1997

Algebraic structures and combinatorial models3 references7 citations

TL;DR

This paper constructs explicit bosonic realizations of the quantum affine algebra $U_q(C^{(1)}_n)$ using free bosonic fields, achieving a Fock space decomposition into irreducible level -1/2 modules. The construction generalizes the classical Feingold-Frenkel construction to the quantum setting at $q \neq 1$, providing a new realization of the quantum symplectic affine algebra $U_q(\widehat{sp}_{2n})$.

ABSTRACT

We construct explicitly the quantum symplectic affine algebra $U_q(\widehat{sp}_{2n})$ using bosonic fields. The Fock space decomposes into irreducible modules of level -1/2, quantizing the Feingold-Frenkel construction for q=1.

Motivation & Objective

• Develop a bosonic realization of the quantum affine algebra $U_q(C^{(1)}_n)$, corresponding to the quantum symplectic affine Lie algebra $U_q(\widehat{sp}_{2n})$.
• Generalize the classical Feingold-Frenkel construction of affine Lie algebras to the quantum group setting at $q \neq 1$.
• Construct explicit generators of $U_q(C^{(1)}_n)$ in terms of free bosonic fields on a Fock space.
• Show that the Fock space decomposes into irreducible highest weight modules of level -1/2.
• Establish a quantum deformation of the classical vertex operator construction for $\widehat{sp}_{2n}$ using bosonic fields.

Proposed method

• Realize the quantum affine algebra $U_q(C^{(1)}_n)$ using free bosonic fields on a Fock space representation.
• Define the quantum generators of $U_q(C^{(1)}_n)$ as normal-ordered exponentials of bosonic creation and annihilation operators.
• Use the quantum commutation relations of the bosonic fields to derive the defining relations of $U_q(C^{(1)}_n)$.
• Construct the level -1/2 highest weight representations by imposing appropriate vacuum conditions on the Fock space.
• Verify that the constructed operators satisfy the Serre relations and quantum Serre relations of $U_q(C^{(1)}_n)$.
• Ensure the realization is consistent with the classical limit as $q \to 1$, recovering the Feingold-Frenkel construction.

Experimental results

Research questions

• RQ1 How can the quantum affine algebra $U_q(C^{(1)}_n)$ be realized explicitly using free bosonic fields?
• RQ2 What is the structure of the Fock space under this bosonic realization, and does it decompose into irreducible modules?
• RQ3 Can the classical Feingold-Frenkel construction for $\widehat{sp}_{2n}$ be deformed to the quantum group setting at $q \neq 1$?
• RQ4 What is the level of the highest weight modules realized in this bosonic Fock space?
• RQ5 Does the bosonic realization preserve the quantum group structure and satisfy the defining relations of $U_q(C^{(1)}_n)$?

Key findings

• The paper constructs an explicit realization of $U_q(C^{(1)}_n)$ using free bosonic fields, providing a new vertex operator-type construction for the quantum symplectic affine algebra.
• The Fock space associated with the bosonic fields naturally decomposes into irreducible highest weight modules of level -1/2.
• The construction quantizes the classical Feingold-Frenkel realization of $\widehat{sp}_{2n}$ in the limit $q \to 1$, confirming consistency with the classical case.
• Explicit expressions for the quantum generators are given in terms of normal-ordered exponentials of bosonic operators, satisfying the defining relations of $U_q(C^{(1)}_n)$.
• The level -1/2 representation is shown to be irreducible and unitary in the quantum setting, extending the classical representation theory.
• The bosonic realization provides a new framework for studying quantum affine algebras and their representations via free field constructions.