### Abstract

Generalized KdV hierarchies associated by Drinfeld-Sokolov reduction with grade 1 regular semisimple elements from non-equivalent Heisenberg subalgebras of a loop algebra G(X)C( lambda , lambda ^{-1}) are studied. The graded Heisenberg subalgebras containing such elements are labelled by the regular conjugacy classes in the Weyl group W(G) of the simple Lie algebra G. A representative w epsilon W(G) of a regular conjugacy class can be lifted to an inner automorphism of G given by w=exp (2i pi ad I_{0}/m), where I _{0} is the defining vector of an sl_{2} subalgebra of G. The grading is then defined by the operator d(m,I_{0})=m lambda (d/d lambda )+ad I_{0} and any grade 1 regular element Lambda from the Heisenberg subalgebra associated with (w) takes the form Lambda =(C_{+}+ lambda C_{-}), where (I_{0}, C_{-})=-(m-1)C_{-} and C_{+} is included in an sl_{2} subalgebra containing I _{0}. The largest eigenvalue of adI_{0} is (m-1) except for some cases in F_{4}, E_{6,7,8}. We explain how these Lie algebraic results follow from known results and apply them to construct integrable systems. If the largest ad I_{0} eigenvalue is (m-1), then using any grade 1 regular element from the Heisenberg subalgebra associated with (w) we can construct a KdV system possessing the standard W-algebra defined by I _{0} as its second Poisson bracket algebra. For G a classical Lie algebra, we derive pseudo-differential Lax operators for those non-principal KdV systems that can be obtained as discrete reductions of KdV systems related to gl_{n}. Non-Abelian Toda systems are also considered.

Original language | English |
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Article number | 016 |

Pages (from-to) | 5843-5882 |

Number of pages | 40 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 28 |

Issue number | 20 |

DOIs | |

Publication status | Published - Dec 1 1995 |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)

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## Cite this

*Journal of Physics A: Mathematical and General*,

*28*(20), 5843-5882. [016]. https://doi.org/10.1088/0305-4470/28/20/016