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Regular Conjugacy Classes in the Weyl Group and Integrable Hierarchies
Generalized KdV hierarchies associated by Drinfeld-Sokolov reduction to grade
one regular semisimple elements from non-equivalent Heisenberg subalgebras of a
loop algebra \G\otimes{\bf C}[\lambda,\lambda^{-1}] are studied. The graded
Heisenberg subalgebras containing such elements are labelled by the regular
conjugacy classes in the Weyl group {\bf W}(\G) of the simple Lie algebra
\G. A representative w\in {\bf W}(\G) of a regular conjugacy class can be
lifted to an inner automorphism of \G given by , where is the defining vector of an subalgebra
of \G.The grading is then defined by the operator and any grade one regular element from the
Heisenberg subalgebra associated to takes the form , where and is included in an
subalgebra containing . The largest eigenvalue of is
except for some cases in , . We explain how these Lie
algebraic results follow from known results and apply them to construct
integrable systems.If the largest eigenvalue is , then
using any grade one regular element from the Heisenberg subalgebra associated
to we can construct a KdV system possessing the standard \W-algebra
defined by 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 . Non-abelian Toda systems are also considered.Comment: 44 pages, ENSLAPP-L-493/94, substantial revision, SWAT-95-77. (use
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