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Continuum analogues of contragredient Lie algebras. (Lie algebras with a Cartan operator and nonlinear dynamical systems). (English) Zbl 0691.17012

This paper presents an axiomatic formulation of continuum analogues of contragredient Lie algebras and gives examples of them. Let E be a vector space over a field \(\Phi\) \((={\mathbb{R}}\) or \({\mathbb{C}})\), K and S be two bilinear mappings on \(E\times E\). The “local part” \(\hat{\mathfrak g}={\mathfrak g}_{-1}\oplus {\mathfrak g}_ 0\oplus {\mathfrak g}_{+1}\) is defined as follows. Each \({\mathfrak g}_ i\) is isomorphic to E as a vector space, \({\mathfrak g}_ i=\{X_ i(\phi)\); \(\phi\in E\}\). Suppose the bracket relations \[ [X_ 0(\phi),X_ o(\psi)]=0,\quad [X_ 0(\phi),X_{\pm 1}(\psi)=\pm X_{\pm 1}(K(\phi,\psi)),\quad [X_{+1}(\phi),X_{- 1}(\psi)]=X_ 0(S(\phi,\psi)) \] hold. Let \({\mathfrak g}'\) be the \({\mathbb{Z}}\)-graded free Lie algebra generated by the local part \(\hat{\mathfrak g}\), and J be the largest homogeneous ideal having a trivial intersection with \({\mathfrak g}_ 0\). The quotient algebra \({\mathfrak g}={\mathfrak g}'/J\) is called a “continuum contragredient Lie algebra” with the local part \(\hat{\mathfrak g}\). In this paper it is assumed that E is a commutative algebra over \(\Phi\) and K,S are given by the \(\Phi\)-linear mappings \(E\to E:\) \[ K(\phi,\psi)=(K\phi)\cdot \psi,\quad S(\phi,\psi)=S(\phi \cdot \psi). \] Examples include the Poisson bracket algebra, continuum limit of certain Kac-Moody Lie algebras, Lie algebras of vector fields on a manifold.
The nonlinear dynamical systems associated via a zero curvature type or Lax type representation, with the continuum contragredient Lie algebras are also considered. These include continuous analogues of the Toda lattices. The authors conclude this paper with some perspectives.
Reviewer: H.Yamada

MSC:

17B65 Infinite-dimensional Lie (super)algebras
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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[1] Saveliev, M.V.: preprint IHEP 88-39, Serpukhov (1988); Integro-differerential non-linear equations and continual Lie algebras. Commun. Math. Phys.121, 283–290 (1989)
[2] Vershik, A.M.: Dokl. Acad. Nauk SSSR199, 1004 (1971)
[3] Feldman, J., Moore, C.: TAMS,234, 289 (1977)
[4] Gel’fand, I.M., Graev, M.I., Vershik, A.M.: Usp. Mat. Nauk28, 83 (1973)
[5] Bogoyavlensky, O.I.: Algebraic contructions of some integrable equations. Izv. Acad. Nauk SSSR, ser. Mat.52, 712 (1988)
[6] Kac, V.G.: Infinite-dimensional Lie algebras. Boston: Birkhäuser 1983 · Zbl 0537.17001
[7] Gel’fand, I.M., Kirillov, A.A.: Publ. Math. INES31, 5 (1966)
[8] Golenisheva-Kutuzova, M.I., Reyman, A.G.: Zap. nauch. Semin. LOMI169, 44 (1988)
[9] Leznov, A.N., Saveliev, M.V.: Group methods for integration of nonlinear dynamical systems. Moscow: Nauka, 1985; Exactly and completely integrable nonlinear dynamical systems. Acta Appl. Math.12, 1–86 (1989)
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