Symplectic and orthogonal Lie algebra technology for bosonic and fermionic oscillator models of integrable systems

TL;DR

This paper develops algebraic tools and explicit L-operators for symplectic and orthogonal Lie algebras, facilitating the study of integrable models with these symmetries using algebraic Bethe ansatz.

Contribution

It introduces a convenient presentation of so(N) and sp(2n) Lie algebras, constructs their metaplectic-type representations, and derives explicit L-operators for integrable systems.

Findings

Explicit L-operators for so(N) and sp(2n) are derived.

Formulas for T operators in rational RTT algebra are provided.

Tools enable efficient algebraic Bethe ansatz analysis.

Abstract

To provide tools, especially L-operators, for use in studies of rational Yang-Baxter algebras and quantum integrable models when the Lie algebras so(N) (b_n, d_n) or sp(2n) (c_n) are the invariance algebras of their R matrices, this paper develops a presentation of these Lie algebras convenient for the context, and derives many properties of the matrices of their defining representations and of the ad-invariant tensors that enter their multiplication laws. Metaplectic-type representations of sp(2n) and so(N) on bosonic and on fermionic Fock spaces respectively are constructed. Concise general expressions (see (5.2) and (5.5) below) for their L-operators are obtained, and used to derive simple formulas for the T operators of the rational RTT algebra of the associated integral systems, thereby enabling their efficient treatment by means of the algebraic Bethe ansatz.