Symplectic and Orthogonal Lie Algebra Technology for Bosonic and Fermionic Oscillator Models of Integrable Systems

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.