Symplectic and orthogonal Lie algebra technology for bosonic and fermionic oscillator models of integrable systems. (English) Zbl 1016.81029
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.
Reviewer: Karl-Hermann Neeb (Darmstadt)
MSC:
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
| 81Q40 | Bethe-Salpeter and other integral equations arising in quantum theory |
| 37K30 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures |
| 35Q40 | PDEs in connection with quantum mechanics |
| 17B80 | Applications of Lie algebras and superalgebras to integrable systems |
| 17B20 | Simple, semisimple, reductive (super)algebras |
Keywords:
symplectic Lie algebra; orthogonal Lie algebra; integrable system; bosonic oscillator model; fermionic oscillator model; metaplectic representation; spin representation; Fock space; Yang-Baxter algebras; quantum integrable models; algebraic Bethe ansatzReferences:
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