Symplectic and orthogonal Lie algebra technology for bosonic and fermionic oscillator models of integrable systems (Q2716687)

To provide tools, especially \(L\)-operators, for use in studies of rational Yang-Baxter algebras and quantum integrable models when the Lie algebras \(so(N)\) or \(sp(2n)\) 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 fermionic Fock spaces respectively are constructed. Concise general expressions for their \(L\)-operators are obtained, and used to derive simple formulas for the \(T\)-operators of the rational RTT algebra of the associated integrable systems, thereby enabling their efficient treatment by means of the algebraic Bethe ansatz.