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A geometric derivation of KdV-type hierarchies from root systems

A geometric derivation of KdV-type hierarchies from root systems,Arthemy V. Kiselev,Johan W. van de Leur

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A geometric derivation of KdV-type hierarchies from root systems   (Citations: 3)
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For the root system of each complex semi-simple Lie algebra of rank two, and for the associated 2D Toda chain $E=\{u_{xy}=\exp(K u)\}$, we calculate the two first integrals of the characteristic equation $D_y(w)=0$ on $E$. Using the integrals, we reconstruct and make coordinate-independent the $(2\times 2)$-matrix operators $\square$ in total derivatives that factor symmetries of the chains. Writing other factorizations that involve the operators $\square$, we obtain pairs of compatible Hamiltonian operators that produce KdV-type hierarchies of symmetries for $\cE$. Having thus reduced the problem to the Hamiltonian case, we calculate the Lie-type brackets, transferred from the commutators of the symmetries in the images of the operators $\square$ onto their domains. With all this, we describe the generators and derive all the commutation relations in the symmetry algebras of the 2D Toda chains, which serve here as an illustration for a much more general algebraic and geometric set-up.
Published in 2009.
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    • ...All notions and constructions from geometry of PDE are standard [2, 9, 13]. We follow the notation of [6, 7, 8]...
    • ...Theorem 5 is illustrated in [8]: for each semi-simple complex Lie algebra of rank two, the Hamiltonian operators ˆ A1 and ˆ...
    • ...Ak. Estimates for the orders of the integrals w for the 2D Toda chains associated with semi-simple complex Lie algebras g were claimed or performed in [5, 8, 11, 14, 15] in various formulations, see Example 1. The upper bound, that the numbers ordx w i − 1 are not greater than...
    • ...of the characteristic Lie algebras (see [11, 14, 15] and also [8])...

    Arthemy V. Kiselev, et al. Symmetry algebras of Lagrangian Liouville-type systems

    • ...The differential orders of w, ¯ w grow as r grows, and the formulas are big already for the Lie algebra G2, see [43, 31]...
    • ...We refer to footnote 22 on p. 51 and to [31] for further comments on this example, which is related to the Boussinesq hierarchy...

    Arthemy V. Kiselev, et al. Involutive distributions of operator-valued evolutionary vector fields

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